Let , , be the roots of the equation t^3 - 5t^2 + 2t + 4 = 0 . For any real number x , let M = bmatrix & 2^x & 3^x 2^ -x & & ( 3 2 )^x 3^ -x & ( 2 3 )^x & bmatrix . Then the value of | adj ( adj M)| is equal to

Step-by-step solution

Given M = bmatrix & 2^x & 3^x 2^ -x & & ( 3 2 )^x 3^ -x & ( 2 3 )^x & bmatrix Expanding the determinant of M along the first row: |M| = ( - ( 3 2 )^x ( 2 3 )^x ) - 2^x ( 2^ -x - 3^ -x ( 3 2 )^x ) + 3^x ( 2^ -x ( 2 3 )^x - 3^ -x ) Simplifying the exponential terms: ( 3 2 )^x ( 2 3 )^x = 1 3^ -x ( 3 2 )^x = 3^ -x 3^x 2^ -x = 2^ -x 2^ -x ( 2 3 )^x = 2^ -x 2^x 3^ -x = 3^ -x Substituting these back: |M| = ( - 1) - 2^x(2^ -x - 2^ -x ) + 3^x(3^ -x - 3^ -x ) |M| = ( - ) - ( - 1) + (1 - ) |M| = - ( + + ) + 2 From the given