For $A = \begin{bmatrix}1 & 2 \\ -1 & 3\end{bmatrix}$, $A^2 - kA + 5I = O$. What is $k$?

Explanation

1. 01

   $A^2 = \begin{bmatrix}-1 & 8 \\ -4 & 7\end{bmatrix}$.

2. 02

   The top-left elements of $A^2 + 5I = kA$ give $-1 + 5 = k$, so $k = 4$; the other three elements agree.