An algebra problem

Algebra Level 4

Let \(D=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}\) and \(P=\begin{pmatrix} 7 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 5 \end{pmatrix}\).

Consider \(A={ P }^{ -1 }DP\). Find \(\det({ A }^{ 2 }+A)\).

\[\] Notation: \( \det(\cdot) \) denotes the determinant function.


Source: Technological Institute of Aeronautics (ITA) admission exam
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