For any $X \ne 0$, the kernel $K$ of $F_X$ has in general a dimension 3 with basis $\{I_3, X, X^2\}.$. Calculus questions and answers. Take a basis of the space formed as the union of a basis of the kernel of $A$ and a basis of the range of $A$. Also, compute B22. For Free. Can all matrices be written as the sum of commuting matrices? If Two Matrices are Similar, then their Determinants are the Same, If 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero Matrix. I was trying to randomly choose $A , B$ and test if they work. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. W= A complementary answer to the answer by user1551. Some of our partners may process your data as a part of their legitimate business interest without asking for consent. In general relativity, why is Earth able to accelerate. Find X and Y, if O A. X = A11A31 OB. You can use decimal fractions or mathematical expressions . Note:- if more than one question uploaded than only first question need to, A: Well answer the first question since the exact one wasnt specified. Note that $U$ depends on $n$ parameters and, roughly speaking, the set of solutions in $U$ is almost a vector space; therefore, the set of solutions in $X$ too. Let X + Y = [8(7&[emailprotected]&5)] For non-singular $A$, $B$ the condition is that they have the same elementary factors. Last modified 07/16/2017. Why doesnt SpaceX sell Raptor engines commercially? Teachoo answers all your questions if you are a Black user! Assume that P is nonsingular and solve this for A. Find all matrices X and Y such that: 2X -, Find the real numbers x and y such that det A =, Your new audit client, Hardwood Lumber Company, has a computerized accounting system, Calculate each of the following from the data in the following table, A company determines that if x thousand dollars are spent on advertising. Then it's useless to diagonalize $A,B$; it suffices to find the solutions with rank $n$ of $(*)$; in a second step, the linear system in $Y$: $XY=A,YX=B$ has a unique solution. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Is there anything called Shallow Learning? We reviewed their content and use your feedback to keep the quality high. First week only $4.99! What does Bell mean by polarization of spin state? The expression to determine the value of the scalar is, A: Topic:- matrix operation Is there liablility if Alice scares Bob and Bob damages something. X = 1/2 [8(10&[emailprotected]&8)] Given the equation: T (x) = A x = b. This websites goal is to encourage people to enjoy Mathematics! Please submit a new question, A: As per guidelines we are supposed to solve first three subdivisions. Must it be true that A=D? $A$ and $B$ have the same characteristic polynomials, CEO Update: Paving the road forward with AI and community at the center, Building a safer community: Announcing our new Code of Conduct, AI/ML Tool examples part 3 - Title-Drafting Assistant, We are graduating the updated button styling for vote arrows. X=A11A311 1 O E. X= - A21A31 OF. post multiply byD+BX. ST is the new administrator. Experts are tested by Chegg as specialists in their subject area. X = A41+A31 OH. What is the first science fiction work to use the determination of sapience as a plot point? Ask your question! X=[1201],Y=[1032],Z=[3412],W=[3241] Find scalars a,b, and c such that W=aX+bY+cZ. Learn more about Stack Overflow the company, and our products. Sample size calculation with no reference, Recovery on an ancient version of my TexStudio file. A marketing team needs to complete a marketing campaign project which involves, When a certain type of thumbtack is flipped, the probability of its. X = [8(5&[emailprotected]&4)] X = ( 0 1 2) Y = ( 1 1 2 1 1 1 2 1 ) where the entries marked by asterisks are arbitrary. Linear Algebra. Y = A21 - A31 Compute B22. Try one of our lessons. For those that are defined, give the size ofthe resulting matrix. it is a hollow matrix) and $X=\operatorname{diag}(x_1,\ldots,x_n)$ has distinct diagonal entries, the equation $XY-YX=A$ gives rise to $(x_i-x_j)y_{ij}=a_{ij}$. Comparing entries, we have the following system of linear equations. Y = [8(7&[emailprotected]&5)] [8(5&[emailprotected]&4)] x + 3 z = x + 2 y y + 3 w = 3 x + 4 y 2 x + 4 z = z + 2 w 2 y + 4 w = 3 z + 4 w. n = tr ( I) = tr ( X Y Y X) = tr ( X Y) tr ( Y X) (by property (1), (2) of the trace) = tr ( X Y) tr ( Y X) (by property (3) of the trace) = 0. Also, compute B22. Y = [8(75&0[emailprotected]1&54)] The identity matrix plays a similar role in operations with matrices as the number 1 1 plays in . Is Spider-Man the only Marvel character that has been represented as multiple non-human characters? 2X + 0 = [8(10&[emailprotected]&8)] What is this object inside my bathtub drain that is causing a blockage? Multiply the second on through by -1 and add it to the first one: x + 2y = [1 3 -2] -x - y = [-2 0 -1]_______________________ y = [-1 3 -3] Then x = [2 0 1] - [-1 3 -3], or x = [3 -3 4] y = [-1 3 -3]. X = 1 Y = Z = 4 W= -1 (a) Find scalars a and b such that z = ax + bY. Choose the correct answer below. Suppose A31 is an invertible matrix. The consent submitted will only be used for data processing originating from this website. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. But there are nonsingular matrices $A$, $B$ which A-1AXD+BX-1=A-1CXD+BX-1=A-1C Choose the correct answer below. One class of solutions that can be easily obtained is given by a diagonal matrix $X$ with distinct diagonal entries. Also given that A must be, A: LHS = (AB)-1 Also, compute B22. $$ This website is no longer maintained by Yu. No If b is an Rm vector, then the image will always be a subspace of Rm. Difference between letting yeast dough rise cold and slowly or warm and quickly, Theoretical Approaches to crack large files encrypted with AES. Find the probability and answer the questions. (a, b, c) =, Consider the following matrices. The list of linear algebra problems is available here. $$ The action of $A$ on that basis is fixed. E.g. How to Diagonalize a Matrix. find 1x3 matrices x and y such that x+2y=[1 3 -2] and x+y=[2 0 1], Jack M. A: Given that the product D=ABCof three square matrices is invertible. X= A21A31 O c. X=A1 A31 O E. X=-A11A31 OD. rev2023.6.2.43474. (, b)%3D (b) Do there exist scalars a and b such that W = ax + bY? Solution. Medium View solution View more For (AB)-1 = B-1A-1 , RHS * inverse of LHS = i, A: Since you have posted multiple sub parts, as per our policy we will answer first three sub parts., A: The Cayley Hamilton theorem states that every square matrix satisfies its characteristic equation., A: It is given thatAXD+BX-1=C. mean? The general problem is: let $A\in M_{m,m},B\in M_{n,n}$. Why does the bool tool remove entire object? Indeed; or any $n$, $ \ F_X(X^n)=[X,X^n]=X^{n+1}-X^{n+1}=0$, therefore $X^n$ belongs to $K$ for any $n$ but because of Cayley-Hamilton theorem, powers above $2$ can be expressed as polynomials in $I,X,X^2$. Z = The blocks of a Jordan form must have size one. What does "Welcome to SeaWorld, kid!" Also, compute B22- [Hint: Compute the product on the left, and set it equal to the right side.] Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. A X = ( 2 a + 3 c 2 b + 3 d a + 4 c b + 4 d) X A = ( 2 a + b 3 a + 4 b 2 c + d 3 c + 4 d) Now you have a system of equations in four variables: 2 a + 3 c = 2 a + b b = 3 c 2 b + 3 d = 3 a + 4 b a + 4 c = 2 c + d Why is it "Gaudeamus igitur, *iuvenes dum* sumus!" Can the logo of TSR help identifying the production time of old Products? X = 1 1 & 0 & 1 \\ $u_iv_i=\lambda_i$. Could entrained air be used to increase rocket efficiency, like a bypass fan? we need to find a, b and c such that A13=24, A12=-19 and A33. Find matrices X | Chegg.com. (Python), Class 12 Computer Science Y = -A41A37 O E. Y=Azzazi? He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo. Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. Let $A:=\begin{bmatrix} 0 & 1 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$. Are there matrices $X,Y\in\mathbb{C}^{n\times n}$ such that $XY=A$ and that $YX=B$? $A$ is the projection onto its range, parallel to its kernel. (d) Find scalars a, b, and c, not all equal to zero, such that ax + bY + cZ = 0. How could a person make a concoction smooth enough to drink and inject without access to a blender? a Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving that if A is an nn symmetric matrix with positive eigenvalues, then there exists a symmetric matrix B such that B2=A. A: Given Math. You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. It only takes a minute to sign up. finding two conditions so that for two polynomials, there exists exactly one matrix, Matrices that Differ only in Diagonal of Decomposition, When does exist matrices T and H such that HCE=TE? How to determine whether symbols are meaningful. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. $$XY - YX = \begin{bmatrix} 0 & 1 & 1 \\ Rearranging the above equationwe, A: Given thatAXD+BX-1=C Fix a natural number $n$ and let $A,B\in\mathbb{C}^{n\times n}$. 2003-2023 Chegg Inc. All rights reserved. O A. X = A11A31 OB. @JosCarlosSantos All the solutions aren't given in that question. 1 & 0 & 1 \\ Here $Ax$ is an element of the range of $A$ and $(I-A)x$ is and element of the kernel of $A$, since $A(I-A)x=(A-A^2)x=0$. x R. Find f (x). Enter your email address to subscribe to this blog and receive notifications of new posts by email. Find matrices X and Y such that the product below has the form indicated. Take a basis of the space formed as the union of a basis of the kernel of A and a basis of the range of A. You'll get a detailed solution from a subject matter expert that helps you learn core concepts. So, my question is: are there necessary and sufficient conditions about $A$ and $B$ so that the problem has a solution? B22 = A22A31' +A21 A32. B22 = A22A31' - A21 OF. Note that the trace of the $n\times n$ identity matrix is $n$: $\tr(I)=n$. Adding (1) and (2) Two similar matrices always have same characteristic polynomials. Choose the correct answer below. No packages or subscriptions, pay only for the time you need. $$, Plus, of course, $\operatorname{tr}(XY-YX) = \operatorname{tr}(XY) - \operatorname{tr}(YX) = \sum_{i,j} x_{ij} y_{ji} - \sum_{i,j} y_{ij} x_{ji} = 0$, so, $A:=\begin{bmatrix} 0 & 1 & 1 \\ 2X - 5Y = [ 1 [1 2] Previous question Next question X = A21A31 O F. X=A11-A3; OH. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. What happens if you've already found the item an old map leads to? If the field is of characteristic zero, there are infinitely many solutions. Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix. See Answer Question: Exercise 2.1.7 Find all matrices X and Y such that: a. Is there a place where adultery is a crime? EA ABT Would the presence of superhumans necessarily lead to giving them authority? Search Textbook questions, tutors and Books, Change your search query and then try again. (Python). | See the answer Do you need an answer to a question different from the above? Choose the correct answer below. Then we take the trace of both sides and obtain. x [1 4 2 5 3 6 ] = [ 7 2 8 4 9 6 ] The matrix given on the R.H.S. Connected: https://math.stackexchange.com/q/1223984. $$ The NS condition is given by a result due to Flanders; cf. 2003-2023 Chegg Inc. All rights reserved. O A. X = A11A31 OB. Could entrained air be used to increase rocket efficiency, like a bypass fan? All possible values of b (given all values of x and a specific matrix for A) is your image (image is what we're finding in this video). The entries on the diagonal from the upper left to the bottom right are all 1 1 's, and all other entries are 0 0. Are there always matrices $X$ and $Y$ such that $XY=A$ and $YX=B$? Your email address will not be published. $XY=A,YX=B$ ? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Need help with something else? Suppose A31 is an invertible matrix. With help of this calculator you can: find the matrix determinant, the rank, raise the matrix to a power, find the sum and the multiplication of matrices, calculate the inverse matrix. Step by Step Explanation. 4 Answers Sorted by: 18 Let's call your matrix A = ( 2 3 1 4) We want a matrix X 2 2 = ( a b c d) such that A X = X A. Describe the set of all matrices C such that $C^{-1}AC = A$, Given $A$, find invertible $B$ such that $B^{-1}AB$ is positive, How do I find all $n n$ matrices $C$ such that $CD = DC.$. Let A,D, and P be nn matrices satisfying AP=PD. Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix? Calculus. The n\times n nn identity matrix, denoted I_n I n, is a matrix with n n rows and n n columns. Would a revenue share voucher be a "security"? Copyright 2023 SolutionInn All Rights Reserved. Class 12 Computer Science Suppose that f is differentiable for all x \in \mathbf {R} x R with f (2) = 3 and f^ {\prime} (x)=0 f (x) = 0 for all x \in \mathbf {R}. Find all matrices X and Y such that: 2X - Find all matrices X and Y such that: 2X - 5Y = [1 2] Find all matrices X and Y such that: 2X - 5Y = [1 2] This problem has been solved! Made with lots of love Seeking a contradiction, we assume that there are matrices X and Y such that X Y Y X = I. Pre-multiply byA-1 What are necessary and sufficient conditions such that $X^2+Y^2$ is invertible? Choose the correct answer below. Algebra Linear Algebra Question Find all matrices X and Y such that: (a) 3X-2Y= [3 -1], (b) 2X-5Y= [ 1 2] Solution Verified Create an account to view solutions Recommended textbook solutions Linear Algebra with Applications 5th Edition ISBN: 9780321796974 (2 more) Otto Bretscher 2,516 solutions Linear Algebra and Its Applications Problems in Mathematics 2020. What are some symptoms that could tell me that my simulation is not running properly? X=\pmatrix{0\\ &1\\ &&2}\Rightarrow Y=\pmatrix{\ast&-1&-\frac12\\ 1&\ast&-1\\ \frac12&1&\ast} OD. If X, Y, A, and B are matrices of the same size, Simplify the following expression where A, B, and C are matrices. "I don't like it when it is rainy." Math 2568 Spring 2017. Let \[A=\begin{bmatrix} 1 & 1 & 2 \\ 2 &2 &4 \\ 2 & 3 & 5 \end{bmatrix}.\] (a) Find Quiz 1. 0 0 Similar questions If A = 310 126 B = 545 411621 then Medium View solution > If AB=111110100 and A+B= 3111518740, then find matrices A and B. We and our partners use data for Personalised ads and content, ad and content measurement, audience insights and product development. Show that if aX+bY+cZ=0, then a=b=c=0. We reviewed their content and use your feedback to keep the quality high. If the algorithm provides an inverse for the original matrix, it is always possible to check your answer. Connect and share knowledge within a single location that is structured and easy to search. rather than "Gaudeamus igitur, *dum iuvenes* sumus!"? Also, compute B22- [Hint: Compute the product on the left, and set it equal to the right side.] It is required to solve for the matrix X such that all matrices arenn, A: To: Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). (My compulsiveness also made me check the other addiitions, but I won't type it here.) Determine Whether Given Matrices are Similar, True or False: If $A, B$ are 2 by 2 Matrices such that $(AB)^2=O$, then $(BA)^2=O$, Trace, Determinant, and Eigenvalue (Harvard University Exam Problem), Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even. RHS = B-1A-1 How can I repair this rotted fence post with footing below ground? In general, the commutator does not uniquely determine the matrices $X,Y$. @copper.hat How would we show that these are indeed all the solutions? (Hint: Compute the product on the left, and set it equal to the right side.) Is it possible for rockets to exist in a world that is only in the early stages of developing jet aircraft? Does the Trace Commute with Matrix Multiplication? Show that there do not exist scalars a and b such that Z=aX+bY. By right distributive law, It only takes a minute to sign up. To view the purposes they believe they have legitimate interest for, or to object to this data processing use the vendor list link below. Find matrices X and Y such that the product below has the form indicated. (c) If ax + bY + cW = 0, is it true that a = b = c = 0? Can anyone help me to find the matrices $X$ and $Y$ such that Why shouldnt I be a skeptic about the Necessitation Rule for alethic modal logics? Conclude. and $B=YX$ for some $X$, $Y$ iff $A$ and $B$ are conjugate in $\text{GL}_2(\Bbb C)$. 3AT+XBT=-2XT+B B22 = A22+ A3;" A32A21 O A. B22 = A22 - A21A31' A32 O c. B22 = A22 - A21 A32A31' O E. B22 = A22 A31' - A21A32 OD. Math. None of the terminology there is known to me. (b) Do there exist scalars a and b such that W = ax + bY? "I don't like it when it is rainy." 1 & 0 & 1 \\ Take any diagonalisable matrix with eigenvalues $0,1$. Is $\tr (A B) = \tr (A) \tr (B) $? Y = A11+A31 O F. Y=A2,A31' OG Y = -A21A31 OH. If you would like to change your settings or withdraw consent at any time, the link to do so is in our privacy policy accessible from our home page.. Y=A11A31 OC. Reciprocally, the matrix $A=PDP^{-1}$ verify $A^2=A$. Most questions answered within 4 hours. Find step-by-step Algebra 2 solutions and your answer to the following textbook question: Find two $2 \times 2$ matrices X and Y, with not all elements equal, such that XY = YX.. . have the same characteristic polynomial but are not conjugate. Let $X'=P^{-1}XQ,Y'=Q^{-1}YP$; then $X'Y'=D,Y'X'=\Delta$ and we may assume that $A=diag(0_{m-n},(\lambda_i)_i),B=diag((\lambda_i)_i)$. To help Teachoo create more content, and view the ad-free version of Teachooo please purchase Teachoo Black subscription. (1) Does the Fool say "There is no God" or "No to God" in Psalm 14:1. O Yes Linear Combination and Linear Independence, Bases and Dimension of Subspaces in $\R^n$, Linear Transformation from $\R^n$ to $\R^m$, Linear Transformation Between Vector Spaces, Introduction to Eigenvalues and Eigenvectors, Eigenvalues and Eigenvectors of Linear Transformations, How to Prove Markovs Inequality and Chebyshevs Inequality, How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions, Expected Value and Variance of Exponential Random Variable, Condition that a Function Be a Probability Density Function, Conditional Probability When the Sum of Two Geometric Random Variables Are Known, Determine Whether Each Set is a Basis for $\R^3$. Experts are tested by Chegg as specialists in their subject area. VS "I don't like it raining.". What maths knowledge is required for a lab-based (molecular and cell biology) PhD? Find matrices X and Y such that the product below has the form indicated. 1 & 1 & 0 \end{bmatrix}$. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Was this answer helpful? b) One solution? Algebra. Therefore $y_{ij}=a_{ij}/(x_i-x_j)$ when $i\ne j$ and $y_{ii}$ is arbitrary. But how do we find all the possible matrices? By assumption, we have $XY-YX=I$. Leave extra cells empty to enter non-square matrices. Find all matrices X such that AX = XA. We and our partners use cookies to Store and/or access information on a device. Yes But this condition is not sufficient: if $A=\operatorname{Id}_n$, then $XY=A\implies YX=A$. I 0 X 0 B12 A11 A12 A21 A22 0 B22 0 0 I A31 A32 B31 . Applications of maximal surfaces in Lorentz spaces, How to make a HUE colour node with cycling colours. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Step by stepSolved in 5 steps with 5 images, A: Given, Displaying ads are our only source of revenue. Could you reference me to a link where I can read more about the polynomial part? This fails for singular $A$, $B$. Solution Verified by Toppr Given A= 312245738 A T=327143258 Let X= 21(A+A T)= 21 312245738 + 21327143258 = 216353885816 = 33/25/23/244 5/248 X T= 33/25/23/244 5/248 =X So X is a symmetric matrix O Yes No (d) Find scalars a, b, and c, not all equal to zero, such that ax + bY + cZ = 0. Your email address will not be published. If we change the equation to: T (x) = A x = 0. (Hint: Compute the product on the left, and set it equal to the right side.) E.g. The best answers are voted up and rise to the top, Not the answer you're looking for? 2X = [8(10&[emailprotected]&8)] X= -A21A31 O G. X = A21+A31 Find Y. All Rights Reserved. I OY A11 A12 0 B12 0 IX A21 A22 = o B22 001 A31 A32 B31 B32 Find X. Find all $n$ by $n$ matrices $A$ such that $A^2 = A.$ [duplicate], What is the square matrix $A$ equal to if $A=A^2$, CEO Update: Paving the road forward with AI and community at the center, Building a safer community: Announcing our new Code of Conduct, AI/ML Tool examples part 3 - Title-Drafting Assistant, We are graduating the updated button styling for vote arrows, Find all invertible $n\times n$ matrices $A$ such that $A^2 + A = 0$, Finding all $\alpha$ such that the linear system has infinite solutions. X + Y + X Y = [8(7+3&[emailprotected]+0&5+3)] Expert Answer 100% (1 rating) Transcribed image text: Exercise 2.1.7 Find all matrices X and Y such that: 3X - 2Y = [ 3 -1 ] b. 1 & 1 & 0 \end{bmatrix}$$. VS "I don't like it raining.". Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Should convert 'k' and 't' sounds to 'g' and 'd' sounds when they follow 's' in a word for pronunciation? OB. 35,000 worksheets, games, and lesson plans, Marketplace for millions of educator-created resources, Spanish-English dictionary, translator, and learning, Diccionario ingls-espaol, traductor y sitio de aprendizaje, a Question Aside from humanoid, what other body builds would be viable for an (intelligence wise) human-like sentient species? Gauss-Jordan Elimination / Homogeneous System. Im waiting for my US passport (am a dual citizen). AX = B AX = B. Math Algebra Consider the following matrices. See A. Clearly, $X$ is a solution of the linear equation $(*)$ $AX=XB$; one can see that the set of solutions of the equation $(*)$ is a vector space of dimension $n$. (, b)%3D (b) Do there exist scalars a and b such that W = ax + bY? Thanks. How can I repair this rotted fence post with footing below ground? (a, b, c) =, Elementary Linear Algebra (MindTap Course List), Algebra & Trigonometry with Analytic Geometry. - 1 -1 O A. X= -A11A31 O B. X = A21A31 O c. X= A21-A31 OD. (i) X + Y = [8(7&[emailprotected]&5)] and X Y = [8(3&[emailprotected]&3)] Definition of identity matrix. Playing a game as it's downloading, how do they do it? I 0 X 0 B12 A11 A12 A21 A22 0 B22 0 0 I A31 A32 B31 B32 Find X. (all matrices are rectangular). Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$, How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix, The Intersection of Two Subspaces is also a Subspace, Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$, Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis, Find a Basis for the Subspace spanned by Five Vectors, Prove a Group is Abelian if $(ab)^2=a^2b^2$, Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space. A: Given Data Set it up and solve it as a system of two linear equations in two variables: x + 2y = [1 3 -2] x + y = [2 0 1]______________________. For a completely rigorous proof, see this answer. For what matrices $B,C$ is there an $A$ such that $AB=C$? The best answers are voted up and rise to the top, Not the answer you're looking for? Learn more about Stack Overflow the company, and our products. Choose the correct answer below. 1. How does one find a set of square matrices that commute with each other? If $A$ and $B$ are non-singular, $X$ and $Y$ have to be too. Y = Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. X = A11A31. Learn more about Stack Overflow the company, and our products. I OY A11 A12 0 B12 0 IX A21 A22 = o B22 001 A31 A32 B31 B32 Find X. b Show that matrices A and B must be identical. A. Albert and Benjamin Muckenhoupt (1957), On matrices of trace zeros, Michigan Math. 4 + x = (2x + y) I get x = 3 2 and y = 5 2. How to show errors in nested JSON in a REST API? Y = [8(2&[emailprotected]&1)] Find matrices X and Y such that the product below has the form indicated. 7 If the field is of characteristic zero, there are infinitely many solutions. Suppose A31 is an invertible matrix. The size of a matrix (which is known as the order of the matrix) is determined by the number of rows and columns in the matrix. 2X - 5Y = [ 1 [1 2] Show transcribed image text Expert Answer 100% (2 ratings) Transcribed image text: Exercise 2.1.7 Find all matrices X and Y such that: a. Observe that $x = Ax + (I-A)x$. Therefore if $(X_0,Y_0)$ is a solution to the equation $[X,Y]=A$, $(X_0,Y_0+aI+bX_0+cX_0^2)$ is as well a solution, for any $a,b,c \in \mathbb{R}$. Is it possible for rockets to exist in a world that is only in the early stages of developing jet aircraft? More generally, over an arbitrary field, it is known that every traceless matrix (including but not limited to hollow matrices) can be written as a commutator. Answer: Undefined Undefined Undefined Undefined Suppose that A, B, C, D, and E are matrices with the following sizes: BC D E In each part, determine whether the given matrix expression is defined. Why are $\left(\begin{smallmatrix}0&1\\0&0\\\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}0&0\\0&1\\\end{smallmatrix}\right)$ not similar? Making educational experiences better for everyone. Therefore, X has to be a 2 2 matrix. Men have XY (or YX . Therefore, the matrix $A$ is determined by saying who is the kernel and a complement of the kernel to be its range. Thus, X = [8(5&[emailprotected]&4)] $$XY - YX = \begin{bmatrix} 0 & 1 & 1 \\ Find matrices X and Y such that the product below has the form indicated. For a trivial example, take $A=I$ and $B=\pmatrix{1&1\\0&1}$. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Find matrices X and Y such that the product below has the form indicated. Connect and share knowledge within a single location that is structured and easy to search. It only takes a minute to sign up. Teachoo gives you a better experience when you're logged in. a001a001a, A: Given: Suppose A31 is an invertible matrix. An example of data being processed may be a unique identifier stored in a cookie. X Y = [8(3&[emailprotected]&3)] Set it up and solve it as a system of two linear equations in two variables: x + 2y = [1 3 -2] x + y = [2 0 1] _____ Multiply the second on through by -1 and add it to the first one: (a) Find scalars a and b such that z = ax + bY. (Hint: Compute the product on the left, and set it equal to the right side.) Necessarily $rank(X)=rank(Y)=n$ and $X,Y$ are in the form $X=\begin{pmatrix}0\\U\end{pmatrix},Y=\begin{pmatrix}0&V\end{pmatrix}$ where the $n\times n$ invertible matrices $U,V$ satisfy $UV=VU=diag((\lambda_i)_i)$. Required fields are marked *. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Consider the following matrices. Connect and share knowledge within a single location that is structured and easy to search. He has been teaching from the past 13 years. Let A = 1 3 3 2 . X = 1 Y = Z = 4 W= -1 (a) Find scalars a and b such that z = ax + bY. How to Find Eigenvalues of a Specific Matrix. Yes No (c) If ax + bY + cW = 0, is it true that a = b = c = 0? From matrix multiplication, we obtain the following system of equations: 3ax1 ax2 = a(3x1 x2) = a (1) 2bx1 +bx2 = b(2x1 + x2) = b (2) 3ax3 ax4 = a(3x3 x4) = 2a (3) 2bx3 +bx4 = b(2x3 + x4) = 2b (4) From (1), we have: 3x1 x2 = 1 x1 = x2 3 1 3 From plugging x1 into (2), we have: 2( x2 3 1 3) + x2 = 5 3x2 2 3 = 1 x2 = 1 x1 = 0 I 0 X 0 B12 A11 A12 A21 A22 0 B22 0 0 I A31 A32 B31 B32 Find X. where the entries marked by asterisks are arbitrary. 2005 - 2023 Wyzant, Inc, a division of IXL Learning - All Rights Reserved, Solving Systems of Equations by Matrix Method. X=\pmatrix{0\\ &1\\ &&2}\Rightarrow Y=\pmatrix{\ast&-1&-\frac12\\ 1&\ast&-1\\ \frac12&1&\ast} Compute $\left[\begin{smallmatrix}1-a & a \\ b & 1-b\end{smallmatrix}\right]^n$, Calculate $\exp \left[\begin{smallmatrix} 4 & 3 \\ -1 & 2 \end{smallmatrix}\right]$, Find two matrices $A$, $B$ such that $A^2 + B^2 = \left[\begin{smallmatrix}2&3\\3&2\end{smallmatrix}\right]$. Suppose A31 is an invertible matrix. I suggest that you edit your answer, adding to it this comment. The matrix is (adsbygoogle = window.adsbygoogle || []).push({}); The Quadratic Integer Ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD), A One-Line Proof that there are Infinitely Many Prime Numbers, Irreducible Polynomial $x^3+9x+6$ and Inverse Element in Field Extension, Difference Between Ring Homomorphisms and Module Homomorphisms, Jewelry Company Quality Test Failure Probability. 1 & 1 & 0 \end{bmatrix}$, Finding matrices $X,Y$ such that $XY - YX = \left[\begin{smallmatrix}0&1&1\\1&0&1\\1&1&0\end{smallmatrix}\right]$, CEO Update: Paving the road forward with AI and community at the center, Building a safer community: Announcing our new Code of Conduct, AI/ML Tool examples part 3 - Title-Drafting Assistant, We are graduating the updated button styling for vote arrows, How find this invertible matrix $C=\left[\begin{smallmatrix} A&B\\ B^T&0 \end{smallmatrix}\right]$, Finding $L\left( \left[\begin{smallmatrix} 3 \\ -1\end{smallmatrix} \right]\right)$, Finding $B,C$ such that $B\left[\begin{smallmatrix}1&2\\4&8\end{smallmatrix}\right]C=\left[\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right]$, Computing the matrix of the linear map $X\rightsquigarrow AXA^t$, where $A=\left[\begin{smallmatrix}2&1\\0&1\end{smallmatrix}\right]$, Finding all $3\times3$ matrices such that $A\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}$ has two distinct solutions. No How many integer matrices have spectral radius bounded by a constant? Y = [8(7&[emailprotected]&5)] X of the equation is a 2 3 matrix. I found it quite interesting. X + Y = [8(7&[emailprotected]&5)] The Vector Space Consisting of All Traceless Diagonal Matrices. The best answers are voted up and rise to the top, Not the answer you're looking for? Diagonalize if Possible. Explanation: Addition of matrices takes place position by position, so we need to solve the additions involving the variables. Hence, X = [8(&@&)] & Y = [8(&@&)]. A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. A necessary condition for that to happen is that $A$ and $B$ have the same characteristic polynomials. I OY A11 A12 0 B12 0 IX A21 A22 = o B22 001 A31 A32 B31 B32 Find X. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. This site uses Akismet to reduce spam. O A. Y = A31A41 OB. Get a free answer to a quick problem. It is a linear operator from the vector space of $3 \times 3$ matrices on itself. Recovery on an ancient version of my TexStudio file, Sample size calculation with no reference. Do there exist $X\in M_{m,n},Y\in M_{n,m}$ s.t. In July 2022, did China have more nuclear weapons than Domino's Pizza locations? Edit : The other question has not given all the matrices. Please login :). 3X - 2Y = [ 3 -1 ] b. Letting X = a b c d , we get 1 3 3 2 a b c d = a b c d 1 3 3 2 , that is, a +3c b +3d 3a +2c 3b +2d = a +3b 3a +2b c +3d 3c +2d . The action of A on that basis is fixed. A=123abc-123 Since n is a positive integer, this is a . Start your trial now! To do so, use the method demonstrated in Example 2.6.1.Check that the products \(AA^{-1}\) and \(A^{-1}A\) both equal the identity matrix. How does TeX know whether to eat this space if its catcode is about to change? Combining these we have $n=tr(XY-YX)$. answered 12/07/12, Keys to Chemistry, Calculus, Literature and More. It isn't too hard to show that there are infinitely many non-invertible matrices, and finitely many invertible ones. 4 + (x y) = 3. X = A21A31 1 1 OG. Manage Settings Remark: Operator $F_X$ could be given a form using Kronecker product. Is there any evidence suggesting or refuting that Russian officials knowingly lied that Russia was not going to attack Ukraine? View solution > If A = 310 126 B = 545 411621 then Medium View solution View more We consider the (generic) case when $m\geq n$ and $B$ is generic (then its eigenvalues $(\lambda_i)$ are non-zero and distinct and $B$ is diagonalizable invertible); thus, NECESSARILY, $spectrum(A)=\{spectrum(B),0_{m-n}\}$ and the eigenvalue $0$ of $A$ is semi simple; then $A=PDP^{-1},B=Q\Delta Q^{-1}$ where $D,\Delta$ are diagonal. rev2023.6.2.43474. My father is ill and booked a flight to see him - can I travel on my other passport? Ex 3.2, 7 (i) - Chapter 3 Class 12 Matrices Last updated at May 29, 2023 by Teachoo Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class 3A+XBT=-2XT+B Here A x is an element of the range of A and ( I A) x is and element of the kernel of A, since A ( I A) x = ( A A 2) x = 0. Putting value of X in (1) Consider the matrices below. Explain how to find A + B, A - B. and matrix C such that A + C is the zero matrix. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. A link to the app was sent to your phone. (2) Thakin Stairs Co. designs and builds factory-made premium wooden stairways for homes. More specifically, when an $n\times n$ matrix $A$ has a zero diagonal (i.e. Save my name, email, and website in this browser for the next time I comment. Living room light switches do not work during warm/hot weather, Citing my unpublished master's thesis in the article that builds on top of it. Is it OK to pray any five decades of the Rosary or do they have to be in the specific set of mysteries? 4 Do you need an answer to a question different from the above? rev2023.6.2.43474. 3X - 2Y = [ 3 -1 ] b. a Find the rank and nullity of AB. Matrix Analysis By Roger A. Horn, Charles R. Johnson is a good reference. Then $A=XY$ Colour composition of Bromine during diffusion? - 1 -1 O A. X= -A11A31 O B. X = A21A31 O c. X=. (b). Let $A=[1 2 0 1 ]$. Two Matrices with the Same Characteristic Polynomial. Notice that $A$ is diagonalizable since the polynomial $x(x-1)$ annihilates it and then $\operatorname{sp}(A)\subset \{0,1\}$ with multiplicity $\dim\ker A$ and $\operatorname{rank}A=:r$ respectively so $A$ is similar to $D=\operatorname{diag}(\underbrace{1,\ldots,1}_{r\,\text{times}},0,\ldots,0)$, i.e. Please repost other questions, A: Take A and B are arbitrary matrices and find AB-BA and analyse the diagonal elements. Why is Bb8 better than Bc7 in this position? Answer Given 2X5Y = [1 2] let Y = T where View the full answer Describe how a system's entropy changes If the system becomes more disordered during a process. Do matrices $ AB $ and $ BA $ have the same minimal and characteristic polynomials? Learn how your comment data is processed. Yes No (c) If ax + bY + cW = 0, is it true that a = b = c = 0? Applications of maximal surfaces in Lorentz spaces. VS "I don't like it raining.". Is there a way to tap Brokers Hideout for mana? Choose the correct answer below. 1 & 0 & 1 \\ Just type matrix elements and click the button. A and B are matrices and k is a scalar. Difference between letting yeast dough rise cold and slowly or warm and quickly, "I don't like it when it is rainy." 2X - 5Y = [1 2] Find all real numbers a and b such that the following system of linear equations has: a) No solutions? A: We are given a matrix: Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. J., 4(1):1-3. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. The elements of the kernel go to zero, the element of the range are fixed. (a, b, c) = O Yes No (d) Find scalars a, b, and c, not all equal to zero, such that ax + bY + cZ = 0. of the equation is a 2 3 matrix and the one given on the L.H.S. How to determine whether symbols are meaningful. @orole edit : I don't understand much after we form the basis from the union. One class of solutions that can be easily obtained is given by a diagonal matrix X with distinct diagonal entries. and caffeine. Is it possible? X= A21A31 O c. X=A1 A31 O E. X=-A11A31 OD. statistics. A: Theorem The elements of the kernel go to zero, the element of the range . Finally $U,V$ are diagonal matrices $U=diag(u_i),V=diag(v_i)$ s.t. Is it possible to type a single quote/paren/etc. Use of Stein's maximal principle in Bourgain's paper on Besicovitch sets. Solution. So we need to solve the system. X= A21A31 O c. X=A1 A31 O E. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Find all matrices B such that A B = B A Since we want to find B such that A B = B A, we need to find x, y, z, w satisfying [ 1 3 2 4] [ x y z w] = [ x y z w] [ 1 3 2 4]. Notify me of follow-up comments by email. X + Y + X Y = [8(7&[emailprotected]&5)] + [8(3&[emailprotected]&3)] Continue with Recommended Cookies, Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class, Ex 3.2, 7 In July 2022, did China have more nuclear weapons than Domino's Pizza locations? (, b)%3D for example the free access paper, https://www.sciencedirect.com/science/article/pii/0024379568900475. Which fighter jet is this, based on the silhouette? Choose an organization for this assignment from one of the following resources. X = [8(10/2&0/[emailprotected]/2&8/2)] Find all $22$ matrices B, $BO_2$ and $BI_2$ such that $AB=BA$. We have to state which property for c(A+B) =cA+cBand(c+d)A=cA+dA, A: Given,AandBbematriceswithsize33det(A-I3)=(-2)(2-9)det(B-I3)=(-3)2(+2)(a)Eigenvalues. when you have Vim mapped to always print two? -1 Also, compute B22- [Hint: Compute the product on the left, and set it equal to the right side.] Choose an expert and meet online.
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