Let A be an n×n matrix. Prove that A and AT have the same Jordan canonical form.

Q: Let A be a m×n matrix. Prove that AB and Ba both exist only if B is an n×m matrix.

A: Explanation of the answer is as follows

Q: Let A be an n× n matrix. Prove that A is a diagonal matrix if and only if Aij= δij Aij for all iand…

A: To prove:

Q: Prove that if A is an n × n invertible matrix, then adj(A−1) = [adj(A)]−1.

A: By definition of A-1, we have

Q: Let A be an n × n matrix. Show that rank A = n if and only if the columns of A are linearly…

A: Show that rank A = n if and only if the columns of A are linearly independent.

Q: Show that if A is a symmetric matrix, then AT is symmetric.

A: we have to show that if A is a symmetric matrix, then AT is symmetric.   as we know that a matrix B…

Q: Prove that the n × n identity matrix is orthogonal.

A:

Q: Show that if A is any mxn matrix, then Im= A.

A:

Q: Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP…

A: Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP…

Q: Use the definition of an inner product to determine if "⟨u, v⟩ = (Au) ̇(Av) for u, v ∈Rn and A is an…

A: Given: ⟨u, v⟩ = (Au)·̇(Av) for u, v ∈Rn and A is an invertible n ×n matrix. To show if <u, v>…

Q: Prove that if A is an invertible n × n matrix, then the rows of A span R“ by using transpose.

A: we use the fact, if A is invertible then A is row equivalent to the identity matrix I.

Q: Prove that if A is a symmetric matrix, then adj A is sym- metric.

A: To Determine:Prove that if A is a symmetric matrix, then adj A is symmetric. that is (adj A)T=(adj…

Q: Let A be an n x n matrix whose characteristic polynomial splits. Provethat A and At have the same…

A: Let A be an n x n matrix whose characteristic polynomial splits. Prove that A and At have the same…

Q: Prove that k(x,x)=xAx' is a valid kernel, where A is a symmetric positive semidefinite matrix.

A: Let  we consider      kx,x' = xTAx'…

Q: Let A be an m x n matrix. Prove that if c is any nonzero scalar, thenrank(cA) = rank(A).

A:

Q: find the Jordan canonical form J for the matrix A,and determine an invertible matrix S such that…

A:

Q: find the Jordan canonical form J for the matrix A,and determine an invertible matrix S such that…

A: Given,

Q: Let A be an m × n matrix with rank m. Prove that there exists a n × m matrix B such that AB = Im:

A: Take any n x m matrix B with rank n. We know that rank AB = rank A = m (for n>m) hence AB is…

Q: Suppose that A is an m×n matrix in which no two columns are identical. Prove that A∗A is a diagonal…

A: Given that A is an m*n matrix in which no two column are identical.Since A*A is defined only if A…

Q: Prove that the n X n identity matrix In is invertible and that

A: We have to prove that the n X n identity matrix In is invertible and that In-1=In.

Q: Let Q be a orthogonal matrix prove that If Q1 and Q2 are orthogonal n X n matrices, then so is Q1…

A:

Q: If A is an n X n matrix, prove that det(adj A) = (det A)n-1

A: Given the matrix A of order n×n, and we have to prove that det(adjA) = (detA)n-1

Q: Prove that if A is a symmetric matrix, then PTAP is also symmetric.

A: Given:           A is a symmetric matrix Prove:           PTAP is also a symmetric matrix In linear…

Q: Show that if A is an orthogonal matrix, then det(A) = ±1.

A: Given If A is an orthogonal matrix. To show det(A) = ±1

Q: Prove: If A and B are m × n matrices such that Ax = Bx for all x ∈ R n , then A = B.

A: Given, If A and B are m×n matrices such that Ax=Bx for all x∈Rn Prove that: A=B

Q: Let A be an m ×n matrix with rank m. Prove that there exists an n ×m matrix B such that AB = Im.

A: Let A be a mxn matrix with rank m. To prove that there exist a nxm matrix B such that AB = Im.

Q: Let Q be an n × n orthogonal matrix. Show that cond2(Q) = 1

A: Q be an n × n orthogonal matrix. To show: ||Q||2=1

Q: Prove: If A is an n × n matrix and Ak = 0 for any k, then An = 0.

A: Let A be a n×n matrixand Ak=0 ∀kWe have to prove An=0Proof-A is a n×n matrixLet λ be an eigen value…

Q: Let A be an n × n matrix. Prove that if Ax = x for all n × 1 matrices x, then A = In.

A: If we have :                     A as an n×n matrix;   To prove:                            If Ax=x…

Q: prove that if A is an invertible matrix then AT is also invertible and (AT)-1= (A-1)T

A: Assume a matrix B. Such that B=A-1

Q: Let A be an n × n matrix. Prove that if Ax = 0 for all n × 1 matrices x, then A = O .

A: we have been given that Ax = 0 for all n × 1 matrix x and  A be an n × n matrix.

Q: Use the SVD to show that any square matrix A can be written as A = unitary and P is Hermitian. UP…

A: By SVD, we know that matrix A can be written as A=SDVTWhere S and V are orthogonal matrices andD is…

Q: Let A be a 4 x 4 matrix. Present systematically all cases for the Jordan Canonical Form of A.

A:

Q: snip

A:

Q: find the Jordan canonical form J for the matrix A,and determine an invertible matrix S such that…

A:

Q: Prove that the n X n identity matrix In is invertible and that ln-l = In

A:

Q: Let A and B be orthogonally similar matrices. Prove that AT and BT are orthogonally similar.

A:

Q: Show that if A is an orthogonal matrix, then det(A) = +1.

A: To prove, detA=±1

Q: Let A be a nonsingular matrix. If det(A)=3, then det(A) = %3D

A: In this question we have to find detA-1.

Q: find the Jordan canonical form J for the matrix A,and determine an invertible matrix S such that…

A: Consider the given matrix and calculate the eigen values and multiplicity factor.

Q: Suppose A is an n×n invertible matrix such that A – 2A = I, . Find an for A. expression

A:

Q: Let A be an invertible matrix. Prove that if A is diagonalizable soi A-1

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Q: Prove that  A + At is symmetric for any square matrix A.

A: Known facts:

Q: Let A be a square matrix such that A2 = O. Prove that (A†)2 = O.

A: Given that A is a square matrix and A2=0.In order to prove (At)2=0 following theorem are…

Q: Let A be an nxn matrix, then for any rER, det(rA) = -det A Select one: True False

A: we have been given a n x n matrix A . we have to determine for every r∈ℝ    , det(rA)=1rdetA

Q: Let A = [a;;] be the n x n matrix defined by aji = k and a;; = 0 if i + j. Show that if B is any n x…

A: Given: A=aijn×n,     aii=k and aij=0 if i≠k. Let B be a square matrix of order 3 which can be…

Q: Prove that |A|=|AT|. Recall, that if A=[aij] and its transpose is AT=B=[bij], then bij=ai

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Q: Show that if A is an orthogonal matrix, then AT is also orthogonal.

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Q: A

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