Question-10

Let \(A=\begin{bmatrix}-1 & 2\\ -4 & 7 \end{bmatrix}\) and \(A^{2}-\alpha A+I=0\) for some \(\alpha\in\mathbb{R}\). Find the value of \(\alpha\).

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We have:

\[ A^{2}=\begin{bmatrix}-1 & 2\\ -4 & 7 \end{bmatrix}\begin{bmatrix}-1 & 2\\ -4 & 7 \end{bmatrix}=\begin{bmatrix}-7 & 12\\ -24 & 41 \end{bmatrix} \]

Now:

\[ \begin{aligned} A^{2}-\alpha A+I & =\begin{bmatrix}-7 & 12\\ -24 & 41 \end{bmatrix}-\alpha\begin{bmatrix}-1 & 2\\ -4 & 7 \end{bmatrix}+\begin{bmatrix}1 & 0\\ 0 & 1 \end{bmatrix}\\ \\ & =\begin{bmatrix}-7+\alpha+1 & 12-2\alpha\\ -24+4\alpha & 41-7\alpha+1 \end{bmatrix}\\ \\ & =\begin{bmatrix}0 & 0\\ 0 & 0 \end{bmatrix} \end{aligned} \]

We can now see that \(\boxed{\alpha=6}\).