Find the inverse, if it exists, of the given matrix. [-6 -6 19) 19) [-5 -5 性 6. 5 11 11 11 11 11 11 A) 6. B) C) 6. D) Does not exist 11 11 11 11 11 11

Q: Prove that if B is the matrix obtained by interchanging the rows of a 2 ×2 matrix A, then det(B) =…

A: Let the matrix A and the new matrix B be as follows.

Q: (AB)† = B†A† (AB)−1 = B−1A−1

A: This is a question of linear algebra. The first one is important property regarding conjugate…

Q: Prove that the following matrix is orthogonal and skew-symmetric: 1 -1 -1 1 1 M = V3 1 1 1 1.

A:

Q: Prove that if A, B, and C are square matrices and ABC = I, then B is invertible and B−1 = CA.

A: If Then Similarly

Q: Compute the adjugate of A = the cofactor matrix of A. -2 0 3 5 -2 -2 1 2 noting that adj(A) =…

A:

Q: Find matrices A and B in M2(Z) such that AB= 0, but BA+0. 3.

A:

Q: Prove that if two rows (columns) of the n × n matrix A are proportional, then det(A) = 0.

A: Consider the n×n matrix A whose two rows consider R1 and R2 are proportional.…

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: For the matrix A in , use the Cayley ­Hamilton Theorem to compute A2, A3, and A4 by expressing each…

A: Given matrix is

Q: Given that "a" is some nonzero constant, calculate the inverse of the following matrix: 5 -За 7 9. 9…

A:

Q: For the matrix A , use the Cayley­Hamilton Theorem to compute A -1 and A -2 by expressing each as a…

A: According to Cayley Hamilton theorem, any square matrix satisfies its own characteristics.

Q: Consider A and B are two square matrices and B is a scalar. Calculate (BA^T B)^(-1) = *

A:

Q: Consider the matrix [1 2 3] B = |2 3 4 3 4 6 (a) Show, by calculating det B, that B has an inverse…

A:

Q: let A nonsingular matrix, c #0, c scalar (cA)" = CA -1 -1 %3D Select one: True O False

A:

Q: State Cayley-Hamilton theorem and prove this for following matrix [1 1 A 1 2 -1 %3D [5 8

A:

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: Given B1 and B2 as the bases of R?. B, = {[",]-C1} and B, = {G-C) %3D Find a. The coordinates of…

A: Ansa; Consider, 10=a11+b3-1⇒10=a+3ba-b⇒a-b=0⇒a+3b=0slove both equation and get;a=b=14 Co-ordinates…

Q: Given a 3x matrix in terms of scalar c and that det (A)=8, find the valuddof G if -1 1c-2 C+7 3. 2.

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

A:

Q: b) Prove: Let A be an m × n matrix, R" = NS(A) RS(A).

A: Given: Let A be an m×n matrix , Rn=NSA⊕RSA

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

A: The objective is to prove that

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: b) Prove: Let A be an m x n matrix, R" = NS(A) + RS(A). %3D

A: A be an m×n matrix, we have to prove that ℝn=NSA⊕RSA  …1.

Q: Find the elementary matrix Ej such that EA = B where 7 A =4 -11 -1 and B = 4 -11 -1 5 6 5 1 12 1 12…

A: We have to solve given problem:

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: Show that if the non-zero vectors s(1), s(2),..., s(m) are conjugate to a positive definite matrix…

A: We are given that the non-zero vectors s(1), s(2), ..., s(n) are conjugate to a positive definite…

Q: Determine whether the given matrix is orthogonal. If it is, find its inverse. in given fig.

A: Determine whether the given matrix is orthogonal. If it is, find its inverse. Let…

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

A:

Q: Let A be the 2 x 3 matrix with entries aij = (i – 1)(j+ 1). Which of the following is the matrix A?

A: A be a 2×3 matrix. This means A2×3 . entries of matrix A are ai,j=(i-1)(j+1)

Q: Calculate adj A, det A, and hence A¯', for the matrix A=| 2 1 -2 -1 1

A: To find the adjoint determinant and inverse of a matrix

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

A: To prove, detA=±1

Q: Show that if a matrix A has an inverse A-, then in fact A-1 has an inverse as well

A:

Q: In each part of Exercises 1-4, determine whether the matrix is orthogonal, and if so find it…

A: Solution is given below:

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

A:

Q: A matrix Q ϵ Mnxn(R ) is called orthogonal if QQt = I . Prove that if Q is orthogonal, then det(Q) =…

A: Given data: The orthogonal matrix is given by "Q". The given condition is  QQT = I.....(1) Now, take…

Q: Prove that if A and B are matrices such that A is symmetric, then (BA−1)T(A-1BT)-1=In

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: Verify that P = is an orthogonal matrix. 1/32/3 N3

A: Given the matrix P=23-23132313-23132323We need to check whether the given matrix is orthogonal…

Q: Show that if the non-zero vectors s1), s(2),..., s) are conjugate to a positive definite matrix G,…

A:

Q: Let A, Band C be square matrices Such that AB = AC, If A#0,then B-C. 1s this True or false? Explain…

A: Let A, B, and C be square matrices such that AB=AC. If A≠0 , then B=C. Determine true or false.

Q: Assume that T and S are matrix of the same size. Prove or Disprove that (T + S) 2 is a symmetric,…

A: Given   T  and   S are matrix of same size. We have to prove if  T+S is symmetric, skew - symmetric…

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: Let B = {b1, b2} and C = {c1, c2} be two bases for R, where %3D bị b2 C1 = C2 = 8 b1 what is the…

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Q: Let A be the n × n matrix given by Show that det (A) = (a - 1) n − 1(a + n - 1)

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Q: If the n x n matrices A and B are orthogonal, which of the matrices in Exercises 5 through 11 must…

A: Given- A and B are n×n orthogonal matrices. To find- A+B is orthogonal or not?     Concept Used-…