Let A = bmatrix 2 & 1 & 1 1 & 2 & 3 & 1 & 2 bmatrix and B = bmatrix 1 2 p bmatrix . If the matrix equation AX = B has infinitely many solutions, then the value of ^2 + p^2 + Tr ( adj (A)) is equal to (where Tr (M) denotes the trace of matrix M and adj (A) denotes the adjoint of matrix A )

Step-by-step solution

For the system AX = B to have infinitely many solutions, we must have |A| = 0 . |A| = vmatrix 2 & 1 & 1 1 & 2 & 3 & 1 & 2 vmatrix = 0 2(4 - ) - 1(2 - 3 ) + 1(1 - 6) = 0 8 - 2 - 2 + 3 - 5 = 0 + 1 = 0 = -1 Substitute = -1 into the system of equations: 2x + y + z = 1 x + 2y - z = 2 3x + y + 2z = p For infinitely many solutions, D_x = D_y = D_z = 0 . Let us check D_y = 0 : D_y = vmatrix 2 & 1 & 1 1 & 2 & -1 3 & p & 2 vmatrix = 0 2(4 + p) - 1(2 + 3) + 1(p - 6) = 0 8 + 2p - 5 + p - 6 = 0 3p - 3 = 0 p = 1 Now, we find the