Matrices & Determinants · Determinant of adjoint

Let

A

be a

3 \times 3

matrix such that

A^{T}\begin{bmatrix} 1\\ 0\\ 1\end{bmatrix} = \begin{bmatrix} 5\\ 2\\ 2\end{bmatrix},\quad A^{T}\begin{bmatrix} 0\\ 0\\ 1\end{bmatrix} = \begin{bmatrix} 3\\ 1\\ 1\end{bmatrix},\quad A\begin{bmatrix} 1\\ 0\\ 1\end{bmatrix} = \begin{bmatrix} 3\\ 4\\ 4\end{bmatrix}\quad\text{and}\quad A\begin{bmatrix} 0\\ 0\\ 1\end{bmatrix} = \begin{bmatrix} 1\\ 3\\ 1\end{bmatrix}

\det(A) = 1

, then

\det(\operatorname{adj}(A^2 + A))

is equal to:

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