Question

If $A = \begin{pmatrix}1&5\\\ 0&2\end{pmatrix}$ , then

• A. $A^2 - 2A + 2I = 0$
• B. $A^2 - 3A + 2I = 0$
• C. $A^2 - 5A + 2I = 0$
• D. $2A^2 - A + I = 0$

Answer 1

Answer: (B)

$A^2 - 3A + 2I = 0$

Answer 2

We have,
$A=\begin{bmatrix}1 & 5 \\\ 0 & 2\end{bmatrix}$
$\therefore A^{2}=A \cdot A=\begin{bmatrix}1 & 5 \\\ 0 & 2\end{bmatrix}\begin{bmatrix}1 & 5 \\\ 0 & 2\end{bmatrix}=\begin{bmatrix}1 & 15 \\\ 0 & 4\end{bmatrix}$
$\therefore A^{2}-3 A+2 I=\begin{bmatrix}1 & 15 \\\ 0 & 4\end{bmatrix}-3\begin{bmatrix}1 & 5 \\\ 0 & 2\end{bmatrix}$
$+2\begin{bmatrix}1 & 0 \\\ 0 & 1\end{bmatrix}=\begin{bmatrix}0 & 0 \\\ 0 & 0\end{bmatrix}$
$\Rightarrow A^{2}-3 A+2 I=0$