Prove that for any m×n matrix A, rank(A) = 0 if and only if A is the zero matrix.

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: Prove that if one row (column) of the n × n matrix A consists entirely of zeros, then A is singular.

A: We know that if all entries of one row(column) of the n×n is zero then determinant of matrix is…

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: Prove that if B is a 3×1 matrix and C is a 1×3 matrix, then the 3×3 matrix BC has rank at most 1.…

A: Let A be the 3 × 3 matrix of rank 1, then it will be in the form as follows.

Q: Let S be an n × n singular matrix. Prove that for any n × n matrix B, the matrix SB is also…

A: Singular matrix :A matrix is singular if and only if it has a determinant of 0

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 for any m x n matrix A, rank(A) = 0 if and only if A is the zero matrix.

A: Given, A is m x n matrix To prove:

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 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 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 and B be nxn inversible matrix. Prove that AB is invertible and (AB)-1=  B-1A-1.

A: Let A and B be n x n invertible matrices, with inverses A-1 and B-1, respectively.

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: Show that A + AT is symmetric for any square matrix A.

A:

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

A:

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

A: Matrix A is said to be non singular if the determinant value is non zero.

Q: Prove that a matrix A is positive definite if and only if it can be written as A = RTR, where R is a…

A:

Q: Suppose that x and y are mxn matrices. Prove that xT- yT=(X -YT. Find a 2x2matrix A, that is not the…

A: Take A and B as arbitrary m×n matrices and evaluate the left and right sided expression

Q: Let A and B be n × n matrices such that AB is invertible. Show that A must be invertible. You cannot…

A:

Q: Prove that, for m X n matrices A and B, rank (A + B) <=rank(A) + rank(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: 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

A:

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: If A be an n x n invertible matrix. Prove that the inverse of A is unique.

A: The assumption for the matrices helps to prove the required. A matrix is said to be invertible if…

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 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: There exists a 3 x 3 orthogonal matrix A such that 2 A 0 = True False

A:

Q: Prove that if A is an invertible matrix then det(A−1) = 1/det(A)

A:

Q: Let A be an n × n matrix with a column of zeros. Show directly from the Leibniz formula that…

A: Let, A be an n×n matrix with a column of zeros.  To prove detA=0 using Leibnitz formula for…

Q: Let A E Mn(C) denote the n x n matrix with all entries equal to 1. Show that A is positive…

A: Let A belongs to  Mn (C) denote the n x n matrix with all entries equal to 1. Show that A is…

Q: PROVE IT Given any real n x n matrix A, if A is invertible, then there is an orthogonal matrix Q…

A:

Q: Prove that if a row or a column of an n × n matrix A consists entirely of zeros, then det(A) = 0.

A:

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: Suppose A is a 2 x 2 matrix such that for all x, y E R², (Ax) · (Ay) = x ·y Must it be true that A…

A: Please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: (4) Let A be an invertible matrix. Prove that if A = Q¸R1 decompositions of A and if the diagonal…

A:

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

A: Known facts:

Q: Show that is A is a square matrix, then A + AT is a symmetric matrix. You can use any matrix for A…

A:

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: 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 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: Prove the following statement: If the columns of A are linearly independent, then AT A is…

A: A is invertible if there exists a matrix A-1 such that AA -1=A -1A=I   The vectors v1,.......,vn are…

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

A:

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

A:

Q: Prove that if B is a 3 x 1 matrix and C is a 1 x 3 matrix, then the 3 x 3 matrix BC has rank at most…

A: Let “B” and “C” be arbitrary 3*1 and 1*3 matrices respectively, B=b1b2b3,   C=c1c2c3 Then the…

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…

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.