Question

1. Let A and B be two 4 by 4 matrices with (let A =-2 and det B-1-8. Find det(-2.1' B) 2. Assume that A is a 4 x 4 matrix and det (Adj(A))-8, find det(A) 3. Find the inverse the given matrice by way of elementary row operations

Answer 1

Given that, Let A and B be two 4 times 4 matrices with det A = -2 and det B^-1 = 8. We have to determine det (-2A^tB). det (-2A^TB) = (-2)^4 det(A^T) det(B) [Because det (alpha) = alpha^n det(A) det (AB) = det(A) det(B)] = 16(2) 1/8 (Because det(A) = det(A^T)] = -4 [Because(A^-1) = 1/(A)] therefore det (-2A^TB) = -4 Assume that A is a 4 times 4 matrix and det (Adj (A) = -8 We know that det(adj(A)) = (det(A))^n - 1 -8 = (det(A))^4 -1 [det(A)]^3 = -8 therefore det(A) = -2 Given matrix A = [1 1 1 1 -1 1 1 0 0] [A/I] = [1 1 1 1 -1 1 1 0 0]/[1 0 0 0 1 0 0 0 1] = [1 0 0 1 -2 0 1 -1 -1/1 -1 -1 0 1 0 0 0 1] R_2 rightarrow R_2 - R_1 R_3