Let A be a 3 3 matrix such that A^T bmatrix 1 0 1 bmatrix = bmatrix 5 2 2 bmatrix , A^T bmatrix 0 0 1 bmatrix = bmatrix 3 1 1 bmatrix , A bmatrix 1 0 1 bmatrix = bmatrix 3 4 4 bmatrix and A bmatrix 0 0 1 bmatrix = bmatrix 1 3 1 bmatrix . If (A) = 1 , then ( adj (A^2 + A)) is equal to:

Step-by-step solution

Let C₁, C₂, C₃ be the columns of A and R₁, R₂, R₃ be the rows of A . From the given equations, we have: A bmatrix 0 0 1 bmatrix = C₃ = bmatrix 1 3 1 bmatrix A bmatrix 1 0 1 bmatrix = C₁ + C₃ = bmatrix 3 4 4 bmatrix C₁ = bmatrix 2 1 3 bmatrix Similarly, for the transpose A^T , the columns correspond to the rows of A : A^T bmatrix 0 0 1 bmatrix = R₃^T = bmatrix 3 1 1 bmatrix R₃ = bmatrix 3 & 1 & 1 bmatrix A^T bmatrix 1 0 1 bmatrix = R₁^T + R₃^T = bmatrix 5 2 2 bmatrix R₁^T = bmatrix 2 1 1 bmatrix R₁ = bmatrix 2 & 1 &