Let A be a diagonal matrix. Prove that A = At.

Q: Let A be an m × n matrix. Show that if x ∈ N(ATA), then Ax is in both R(A) and N(AT ).

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Q: Given a square matrix A, prove that A is invertible if and only if ATA is invertible.

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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: Let A be an m×n matrix. Show that if A has linearly independent column vectors, then N(A) = {0}.

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Q: Show that if A is an upper triangular matrix, then A T is lower triangular.

A: A is an upper triangular matrix. That is, Now let A=aij be a n×n upper triangular matrix. Then,…

Q: Prove the given theorem: For any square matrix A with real number entries, A + A′ is a…

A: Given: Prove that for any square matrix A with real number entries, A + A′ is a symmetricmatrix and…

Q: Let A be an invertible matrix and let Az = b (A+AA)(x+ Ar) = b. inequality ||AT|| A

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Q: Assuming that b # 0, find a matrix that orthogonally diagonalizes a= [: :] P

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Q: Prove that the rank of a matrix, every element of which is unity is 1

A: Rank of a matrix refers to the number of linearly independent rows or columns of the matrix.To find…

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

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Q: Let A be a nonsingular 4 x 4 matrix such that A = -2, then AT adj(A ) Select one: O True O False 17

A: properties of the determinant

Q: Find x such that the matrix A is nonsingular.

A: Given: The inverse of a matrix is given as follows,

Q: Prove that if A is a nonsingular upper triangular matrix, then A−1 is upper triangular.

A: we have that A is a nonsingular upper triangular matrix and A-1 be its inverse.

Q: Let A be a 3 x 3 matrix such that det(A) = 2. Evaluate det(Aª) and det(2A).

A: Recall the following. If A is a n×n matrix and m be any positive scalardet k A=kndet Adet Am=det Am

Q: Show that for an invertible matrix A: 1 det (A-1) det (A)

A: Given that, the matrix A is invertible. It is known that, detI=1 and detA·A-1=detI.

Q: Show that if A is any m × n matrix, then AAT and AT A are symmetric.

A: There is a matrix  A given of order m×n. To show that AAT  and  ATA  are symmetric. Solution: The…

Q: Prove that B is row equivalent to A if and only if there exists a nonsingular matrix M such that B =…

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Q: Prove that for any matrix A ∈ Mm×n(F), (R(LA∗))⊥= N(LA).

A: The objective is to prove that

Q: Prove If A is an invertible matrix, then AT is invertible and (AT)-1=(A-1)T

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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: Suppose A E M,xn(C) is a normal, invertible matrix. Prove that A A- is unitary.

A: Given A € Mnxn(C) is a normal matrix.

Q: Let A be an m × n matrix. Prove that if AT A is nonsingu- lar, then rank A = n.

A: Given that, A is an n x n matrix. A^T is the transpose matrix of A. AT A is non-singular. We have to…

Q: Let A be an orthogonal matrix. Show that cA is orthogo- nal if and only if c = ±1.

A: Given:            A be an orthogonal matrix.

Q: Let A be an nxn matrix. If A-= det A adj A is computed, what should AA be equal to in order to…

A: Using the given matrix equation we need to find the value of AA-1 .

Q: Let B be an invertible matrix. Show that A = BTB is positive definite.

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Q: Let A be an Hermitian matrix and let B = iA. Show that B is skew Hermitian.

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Q: let B- 8. find a triangel ar matrix A such that

A: A triangular matrix is a type of square matrix. These are of two types:  Lower triangular  upper…

Q: Show that if A is an m × n matrix such that AAT is non- singular, then rank A = m.

A: Let AAT can be non singular only if the columns in A are linear independent and they span the column…

Q: Let A and B be n x n matrices, and let c be a scalar. Prove that(A + cB )* =A* + cB*.

A: Consider the given relation, A+cB*=A*+cB* The Conjugate transpose of n×n matrix A is n×n matrix A*…

Q: Let A be a square matrix such that A2 = 0. Prove that (Af)2 = 0

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

Q: Let A be a 2 x 2 matrix. If det (A) # 0, then det (adj A) = det (A).

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Q: Let A = UEVT be a SVD of a square matrix A. What is the product of the diagonal elements of E, in…

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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: Show that if A is an n x n symmetric matrix, then |(Ax)•y = x•(Ay) for all x, y in R".

A: In symmetric matrix, transpose of the matrix is equal to the matrix. AT=A

Q: Let A be a nonsingular 4 x 4 matrix such that |A-1| = -2, then AT.adj(A-}) Select one: O True O…

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Q: Let A be an n x n matrix. If vi, v2,..., v are vectors in R" such that {Avi, Au2,..., Avn} is…

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Q: 16. Prove that for any square matrix A, ||A||ž E ||A* A||2.

A: Given: A is a square matrix.

Q: Let A be a nonsingular 4 x 4 matrix such that A = -2, then adj(A) - Select one: True False

A:

Q: Please see image

A: Given matrices are

Q: Prove that if A is an invertible matrix and B is row equivalentto A, then B is also invertible.

A: Recall the following equivalent statements as follows.

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 be a nilpotent matrix (that is, Am = 0 for some m > 1). Prove that if A is diagonalizable,…

A: Consider a nilpotent matrix A. So Am=0 for some m>1.

Q: Let A be an m × n matrix.(a) Describe the diagonal entries of AT A in terms of the columns of A.(b)…

A: (a) Describe the diagonal entries of AT A in terms of the columns of A.

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: show that if At exists for matrix A > then A is to

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

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Q: Let A = [aij] be a square matrix. a. If A has a zero row (column), then prove that det A = 0.

A: Let A = [aij] be a square matrix and If A has a zero row (column). We need to prove that det A = 0.…

Q: Prove that if A is an orthogonal matrix, then so are AT and A−1.

A: We have to prove that if A is an orthogonal matrix, then so are AT and A−1. If A is an orthogonal…

Q: Let A be an n xn Hermitian matrix and let æ be a vector in C". Show that if c= xAx, then c is real.

A: