If A= [ array rrr 1 & -1 & 1 0 & 2 & -3 2 & 1 & 0 array ], B= adj A and C=5 A , then | adj B| |C| =

Step-by-step solution

Calculating the required ratio involves using determinant properties of adjoint matrices and scalar multiplication. The determinant of matrix A = bmatrix 1 & -1 & 1 0 & 2 & -3 2 & 1 & 0 bmatrix is |A| = 1(0 + 3) + 1(0 + 6) + 1(0 - 4) = 3 + 6 - 4 = 5 . Since B = adj A and A is 3 3 , the determinant property | adj M| = |M|^ n-1 gives |B| = |A|^2 = 25 . The adjoint of B similarly yields | adj B| = |B|^2 = 625 . For matrix C = 5A , the scalar multiplication property |kM| = k^n|M| gives |C| = 5^3 5 = 625 . | adj B| |C|