Let A = bmatrix 2k-1 & 1 & 1 0 & 2k-1 & 1 0 & 0 & 2k-1 bmatrix and B = bmatrix 0 & 2k-1 & 1 1-2k & 0 & k -1 & -k & 0 bmatrix where k is a real number. If ( adj A) + ( adj B) = 11^6 , then the value of k - 5 is equal to...

Step-by-step solution

The determinant of the upper triangular matrix A is the product of its diagonal elements: |A| = (2k-1)^3 The matrix B is a skew-symmetric matrix of odd order ( 3 3 ), so its determinant is zero: |B| = 0 Using the property | adj M| = |M|^ n-1 for a matrix of order n , we have: | adj A| = |A|^2 = ((2k-1)^3)^2 = (2k-1)^6 | adj B| = |B|^2 = 0 Given | adj A| + | adj B| = 11^6 , we get: (2k-1)^6 + 0 = 11^6 2k-1 = 11 2k = 12 k = 6 2k = -10 k = -5 Checking the given options for the value of k - 5 : If k = 6 , k - 5 = 6 - 5