4) The Cayley-Hamilton Theorem states that every square matrix satisfies its characteristic equation; that is, if M is an nxn matrix whose characteristic equation is 2" +g2"+...c2+c, = 0, then n-1 M" +c,M" +..c,M +c, =0.Verify the Cayley-Hamilton Theorem for the following matrix. [1 A -B] M =0 -1 0 0 2 C

Q: Suppose A = PRP–1, where P is orthogonal and R is upper triangular. Show that if A is symmetric,…

A: According to the given information suppose, It is required to show that if A is symmetric then R is…

Q: 2. Regarding DFT matrix W₁, prove that W 1 = W for N=4. N

A: Consider the given equation, WN-1=1NWNH for N=4 Where, WN is a DFT matrix. So, as we know that,…

Q: a) If a 7 × 5 Matrix A has rank 2, find dim(Null A), dim (Row A) and rank AT

A: 1) Given  7×5 matrix  rank(A)=2 using rank nullity theorem we have rank A + null A= number of…

Q: Consider the 6 × 8 matrix 3 0 0 0 0 1 2 1 0 0 0 0 1 0 0 -3 4 -7 -9 A = 0 0 0 0 0 0 0 0 0 1 8 1 and…

A: The main objective is to answer the sub parts by using given matrix which is in reduced row echelon…

Q: , Does there exist an 8 x 8 matrix A = (as) satisfying the following three conditions? (i) If i 2 j…

A:

Q: 2.3 Compute the principal square root of each of the llowing positive semidefinite matrices. (b) 2 3…

A: Please find the solution in next step

Q: 6. If A is an nxn matrix whose LU decomposition is PA = LU, , prove that det(A) = (-1)' u₁₁ U 22.²…

A:

Q: Let A = XYT, where X is an m × r matrix, YT is an r × n matrix, and XTX and YTY are both…

A:

Q: If the kernel of a 5 x 4 matrix A consists of the zero vector only and if Au = Aw for two vectors v…

A: It is an necessary and sufficient condition for one one function

Q: [3 Compute the natrix exponential e4t for the diagonal matrix A = !!

A:

Q: Does there exist an 8 x 8 matrix A - (a) satisfying the following three conditions? (1) If i 2j then…

A:

Q: 2. Suppose A is a square matrix. Which matrix is not symmetric? O A+AT O AAT ОА-АТ O ATA

A: Note the following results if A and B are two matrices ABT=BTAT…

Q: a) Show that there is no matrix A E M250x250 with real entries satisfying AAT = -I.

A:

Q: 6. Show that if Q is an orthogonal matrix, then 1 0 %3D 0 Q is also an orthogonal matrix.

A: We have to show that HHt = I

Q: 5. Find a dlagonalizes the fouowing natrlx A, and determine the diagonal matrix D = p'AP matrix p…

A:

Q: 4. Prove that if A is a square matrix, then det(A™ A) = det(AA") %3D

A:

Q: 2. Suppose that A is a 64 × 17 matrix and that the rank of A is 11. What is the dimension of the…

A: The given matrix is A of size 64×17. Also the rank of A is 11. As we know that for any matrix of…

Q: Prove that if A is an invertible matrix and B is row equiva- to A, then B is also invertible. ation

A:

Q: A matrix A is similar to matrix B, if for some invertible matrix P, B = P''AP. Is B similar to A?

A:

Q: Let A be a m times nm×n matrix such that Ax​=b and Ax​=c each have a solution for some choice of b…

A: Let A be an m×n matrix such that Ax​=b and Ax​=c each have a solution for some choice of b and c in…

Q: 1. Find a condition on a, b, c, d such that the augmented matrix " E b has a unique solution.

A: The condition on a,b,c,d must be a relation between a,b,c,d.

Q: Find the characteristic equation for the matrix 2 (13). and use it to compute M-¹. M

A:

Q: Suppose that A is a 5 x 5 matrix which has a null space of dimension 4. The rank of A is rank(A) =

A:

Q: 4. Suppose A is a nonsingular matrix with SVD A= UEV*.

A:

Q: Suppose that A € Mnxn(R) is a positive-definite symmetric matrix. Since A is symmetric, A is or-…

A: We will use the knowledge of standard value decomposition (SVD) of a square matrix.

Q: If A is a 6 × 8 matrix, what is the smallest possible dimension of Nul A?

A: Consider the provided matrix, Given, A is a 6 x 8 matrix. Since, the rank of A equals the number of…

Q: [1 1 Show that the matrix A is not diagonalizable.

A:

Q: 2 Compute the inverse of the matrix A = -2 1 0. . Type DNE if A is not 3. invertible.

A:

Q: Let E be the elementary 3 x 3 matrix corresponding to the elementary row operation of multiplying…

A: Elementary matrices: Elementary matrices are n×n square matrix which is obtained performing…

Q: Let Q be an m xn matrix with n >n and orthonormal columns, then we have both QTQ = I, and QQT = Im…

A:

Q: The Cayley-Hamilton Theorem states that every square matrix satisfies its characteristic equation;…

A: M=1A-B0-1C002 Now characteristic equation is, M-λI=0⇒1-λA-B0-1-λC002-λ=0⇒1-λ-1-λ2-λ-0=0    By…

Q: 1 Problem 2. For a nonsingular matrix A and nonzero scalar ß, show that (BA)- =A-. Problem 3. Prove…

A:

Q: 2 4. Find the matrix A satisfying A and A 1 and write the SVD of A in vector form. [Hint: Start by…

A:

Q: In all parts, suppose that A is a matrix of size 11 x 8 (i) Assuming that rank(Atr) = 3, determine…

A: It is given that

Q: 6. Prove that the matrix A = -1 2 is diagonalizable, and find the corresponding diagonal matrix D…

A:

Q: The Cayley-Hamilton Theorem states that every square matrix satisfies its characteristic equation;…

A:

Q: d the third column of A 'without computi 1 be found without computing the other om nted matrix (A…

A: The matrix is given as A=-2-3-2525431218. To determine the third column of A-1 without computing the…

Q: Suppose A is a 4 x 4 matrix and B is a 4 x 2 matrix, and let uo, ..., U3 represent a sequence of…

A:

Q: If the rows of an mxn matrix A are linearly independent and Ax=b is consistent then it has unique…

A:

Q: 3) A = [3 Calculate the 10th power of the matrix with the Cayley-Hamilton Theorem.

A: Given:

Q: 7. Show that the matrix A=I– 2vv' , where v is a unit vector in R", and I is an nxn identity matrix,…

A:

Q: If an nxn matrix A is invertible, then the rowa of A' are linearly independent. T O True O False

A: If a matrix is invertible then its rows are linearly independent

Q: Is it true that when an n*n matrix whose columns form a basis of R^n, then A is invertible?

A: We shall answer this in next step

Q: 4. Find a diagonal matrix A that satisfies the given condition. 16 At = 0 81 0 1. 01

A:

Q: Suppose that A e Mnxn(R) is a positive-definite symmetric matrix. Since A is symmetric, A is or-…

A: We will use the knowledge of standard value decomposition.

Q: 2. Using Gauss-Seidel, compute a solution to the problem Ax=b where A is an nxn matrix of the form A…

A:

Q: Let A be an n × n matrixSuppose that the matrix C is obtained from A by interchanging the ith and…

A: Matrix C is obtained from A by interchanging the ith and jth row of A. We need to find an elementary…

Q: Let A be an m x n matrix. Prove that if B can be obtained from A by anelementary row [column]…

A:

Q: Suppose that A is a 4 x 4 matrix such that det A = 10. Enter the value of det(A³).

A:

Q: 1) Find a diagonal matrix A that satisfies A-2 20,

A: Here, we need to find a diagonal matrix A such that A-2=2004

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.