Let M be a matrix of integer coefficients. Prove that M-1 exists and has integer coefficients if and only if det M = +1. Let {21,..., zn} be any basis of Z". Let M be an n xn matrix of integer coefficients, with

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 R is a matrix in echelon form, then a basis for row(R) consists of the nonzero rows of…

A:

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

A:

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

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

Q: Verify that: : is an orthogonal matrix.

A: Given matrix, P=23-23132313-23132323

Q: Find an orthonormal basis for the column space of C, a 5(row)x4(column) matrix. C = (-2 -1 5 -3)…

A:

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

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: Let Q be an orthogonal matrix Show that | det(Q)| = 1.

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: Let A be an m ×n matrix and B be an n ×p matrix. Prove that AB can be written as a sum of n matrices…

A: Given that A be a mxn matrix and B be a nxp matrix.Now, prove that AB can be written as a sum of n…

Q: Let x1, x2, and x3 be linearly independent vectors in R4 and let A be a nonsingular 4 × 4 matrix.…

A: Given that:

Q: Let A be an mxn-matrix. Which of the following statements is necessarily TRUE? A. The columns of A…

A: Let's look at each of the options. The statements are straight forward.

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

A:

Q: Let A be a nonsingular n×n matrix with a nonzero cofactor Ann, and set c = det(A)/Ann Show that if…

A: Let A is any non singular nxn matrix with non zeroCofactor Ann and Set C = det(A)Ann   By concept…

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: Find a basis and dimension for the all 4 x 4 skew symmetric matrices with a11 = a12 a9 over R.

A:

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: 3. Suppose A E Mnxn(C) is a normal, invertible matrix. Prove that A* A-1 is unitary.

A:

Q: Prove If U is a unit upper triangular matrix, then U is nonsingular and U−1 is also unit upper…

A:

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: 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: Find a basis for the nullspace of the matrix. (If there is no basis, enter NONE in any single cell.)…

A:

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: Find a basis of Null(A), C(A), and C(AT) for each of the following matrices [1 2] 3 4 (i) A 1 2]…

A:

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: 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: Let U be an orthogonal matrix. Prove that det(U) is either 1 or –1.

A: Orthogonal Matrix: A matrix A is said to be an orthogonal matrix if AAT=I where I  is the identity…

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: A matrix A is orthogonal if A = A' . Prove that for an orthogonal matrix A we have det(A) =±l

A: Given that: For orthogonal matrix, A-1=AT   To Prove: det(A)=±1

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

A:

Q: Find the nullity, rank and a basis for the null space of the matrix A, where (1 3 3 A =| 2 4 -4 1 1

A: first we will find the reduced row Echelon form of the given matrix (rref(A)). Rank of A is number…

Q: Let A = CD, where C and D are n × n matrices and C is invertible. Prove that the matrix product DC…

A: A = CD, where C and D are n × n matrices, and C is invertible. We need to Prove that the matrix…

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

A:

Q: 3. Prove that (i) /, =1, and (ii) ||, =Vn (I is the Identity Matrix of order n).

A:

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

A:

Q: 2. Evaluate: 0020 2020 10 002 0022 Do columns of this matrix form a basis of R4? .

A: Columns of this matrix will form a basis of R4 only if columns forms a linearly independent set…

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 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: × 4 matrix such that

A: In the given question, the concept of reduced row echelon form is applied. Reduced Row Echelon Form…

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

A:

Q: Prove that if A is an orthogonal matrix, then ∣A∣ = ±1.

A: Given: A is orthogonal matrix   To Show: A=±1

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…