If A = [(2 3), (1 2)], verify that A^2 – 4 A + I = O, where I = [(1 0), (0 1)] and O = [(0 0), (0 0)] . Hence, find A^–1

Question

If \(A=\begin{bmatrix}2&3\\1&2\end{bmatrix}\), verify that A² – 4 A + I = O, where \(I=\begin{bmatrix}1&0\\0&1\end{bmatrix}\) and \(O=\begin{bmatrix}0&0\\0&0\end{bmatrix}\). 

Hence, find A ^–1.

Answer 1

A² = \(\begin{bmatrix}2&3\\1&2\end{bmatrix}\)\(\begin{bmatrix}2&3\\1&2\end{bmatrix}\)\(=\begin{bmatrix}4+3&6+6\\2+2&3+4\end{bmatrix}\)

\(=\begin{bmatrix}7&12\\4&7\end{bmatrix}\)

4A = \(4\begin{bmatrix}2&3\\1&2\end{bmatrix}\)\(=\begin{bmatrix}8&12\\4&8\end{bmatrix}\)

\(I=\begin{bmatrix}1&0\\0&1\end{bmatrix}\)

Now, A² – 4 A + I \(=\begin{bmatrix}7&12\\4&7\end{bmatrix}\)\(-\begin{bmatrix}8&12\\4&8\end{bmatrix}\)