Question

For $\lambda \in R, $ let $f(\lambda)=det(A-\lambda I)$ where $A=\begin{bmatrix}1 & 2 \\ -1 & 3\end{bmatrix}$ and $I$ is an identity matrix of order $2$ . The minimum value of $f(\lambda)$ is equal to

Answer 1

We have $f(\lambda)=det(A-\lambda I)=\begin{bmatrix}1-\lambda & 2 \\ -1 & 3-\lambda\end{bmatrix}$
$=(1-\lambda)(3-\lambda)+2$
$=\lambda^{2}-4 \lambda+5=(\lambda-2)^{2}+1$
Clearly, $f_{\min }(\lambda=2)=1$