Show that if A exists for an n x n matrix A, then A- is unique.

Q: Prove that for an n×n matrix A, if A^2 is invertible, then A is invertible.

A: Since, if a matrix A is non-zero then A is invertible.

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: Find x such that the matrix is singular. A = -3 -2 X =

A: Given A = 6x-3-2

Q: Show that if A is any m × n matrix, then Im A = A and AIn = A.

A: We know that if A is a matrix of order m x n and B is another matrix of order n x p, then their…

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: Prove that if A is an n × n matrix, then A − AT is skew-symmetric.

A: Let  A be an n × n matrix, To prove : A-AT is skew-symmetric, that is (A-AT)T=-(A − AT)

Q: Show that the matrix A given below is singular. A = -2

A: A matrix is singular if it's determinant is zero.

Q: IF A is a nonzero n X n matrix such that A 0, then A is singular. %3D Select one: True False

A:

Q: Prove that if A is an n X n matrix, then A - Ar is skew-symmetric

A:

Q: Prove that if an n × n matrix A is not invertible, then A[adj(A)] is the zero matrix.

A: 1). Matrix A is invertible if and only if matrix A is a non singular matrix. That is, determinant of…

Q: Prove that a 2 X 2 matrix A is reducible. if and only if a12=0 and a21=0

A:

Q: Let A, B, and C be three nonzero n × n matrices such that AC = BC. Does it follow that A = B? If so,…

A:

Q: Give an example of a nonzero 2 X 2 matrix A such that A2=O

A:

Q: Prove that if A is a nonsingular diagonalizable matrix, then A−1 is also diagonalizable.

A: Suppose A is non-singular then diagonalizable matrix. It implies that there exists an invertible…

Q: Show that A = 3 and B = are similar matrices by finding an invertible matrix P satisfying A = P-'BP.…

A: We are given two matrices A and B and we have to show that these are similar matrices.For this we…

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: Suppose A is a 2 x 2 matrix such that A and 1 A What is A-1?

A:

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: Prove that if A and Bare symmetric n X n matrices, then so is A + B

A:

Q: Show that if A is a symmetric nonsingular matrix then A−1 is also symmetric.

A: A is a symmetric non singular matrix, which means inverse exists and

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

A:

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: 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: If A is a square matrix such that A =1, then A'is equal

A: Given, A is a square matrix such that A2=I.

Q: If A is a square Matrix such that A'-A, what are the possible values of det A?

A: Determinant possible values

Q: If A is an n × n matrix, show that (Ak )T = (AT )k for any positive integer k.

A: Matrix: We will solve it by the method of induction:(Ak)T=(AT)k case 1: k=1case 2: k=ncase 3: k=n+1

Q: Let A ∈ Mn × n (R) such that A3 = In × n and A≠In × n Give an example of a 3 × 3 matrix that…

A:

Q: If A is an invertible n X n matrix, show that adj A is also invertible

A: Given: To Prove: Concept: 1. 2.

Q: Show that if A is a symmetric positive definite matrix, then A is nonsingular and A−1 is also…

A:

Q: Let A be a m xn matrix with columns c1, ... , en. Prove that {c1, ..., cn} is independent if and…

A: To prove  c1,c2,...cn is independent if and only if null A=0

Q: Find a 3 x 3 matrix A such that A A A A = 0 0 = 2 5 0 3 and

A: Given A100=-1-41,  A010=251  and A001=003

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: snip

A:

Q: IF A is a nonzero n X n matrix such that A = 0, then A is singular. Select one: O True False

A:

Q: Show that if A is any m × n matrix, then Im A = A and A In = A.

A: Given: Matrix A of order m × n.

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: An n x n matrix A is invertible if ВА In АВ II

A: Inverse Matrix: Let A be an n × n matrix. If there exists an n × n matrix B such that AB = I = BA,…

Q: Find a non-zero 2 × 2 matrix A such that A2 = A =

A: We have to find a A matrix of 2x2 such that, A2  = 0000  So let suppose a matrix    Satisfy the…

Q: Show that for any n x n matrix A there exists another n x n matrix M such that MAM-1 = AT, where AT…

A: Consider the given information. Let the matrix A is an n×n matrix and M is a m×m matrix. And the…

Q: Prove that A+ A' is a symmetric matrix, w

A:

Q: snip

A: If a square matrix A of order n is invertible then |A|≠0

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

A:

Q: Find a 3 x 3 matrix A such that Aa = 9x for all a in R³

A: We have to find matrix A such that Ax = 9x

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

A: Consider the given matrix n×m with rank m

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

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 A is any n × n matrix, then (a) A + AT is symmetric. (b) A − AT is skew symmetric.

A: Symmetric matrix: Any matrix A is symmetric matrix if and only if it is equal to its transpose that…

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

A:

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.