If A = [(3 1), (-1 2)], show that A^2 – 5A + 7I = O. Hence, find A^–1.

Question

If \(A=\begin{bmatrix}3&1\\-1&2\end{bmatrix}\), show that A² – 5A + 7I = O. 

Hence, find A ^–1.

Answer 1

\(A=\begin{bmatrix}3&1\\-1&2\end{bmatrix}\)

A² =\(\begin{bmatrix}3&1\\-1&2\end{bmatrix}\)\(\begin{bmatrix}3&1\\-1&2\end{bmatrix}\)\(=\begin{bmatrix}9-1&3+2\\-3-2&-1+4\end{bmatrix}\)

\(=\begin{bmatrix}8&5\\-5&3\end{bmatrix}\)

Now, A² – 5A + 7I\(=\begin{bmatrix}8&5\\-5&3\end{bmatrix}\)\(-5\begin{bmatrix}3&1\\-1&2\end{bmatrix}\)\(+7\begin{bmatrix}1&0\\0&1\end{bmatrix}\)