Question

Show that the matrix \[A = \begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix}\]  is  root of the equation A² − 12A − I = O

Answer 1

\[Given: A = \begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix}\]
\[\]
\[Now, \]
\[ A^2 = AA\]
\[ \Rightarrow A^2 = \begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix}\begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}25 + 36 & 15 + 21 \\ 60 + 84 & 36 + 49\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}61 & 36 \\ 144 & 85\end{bmatrix}\]
\[\]
\[ A^2 - 12A - I\]
\[ \Rightarrow A^2 - 12A - I = \begin{bmatrix}61 & 36 \\ 144 & 85\end{bmatrix} - 12\begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix} - \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow A^2 - 12A - I = \begin{bmatrix}61 & 36 \\ 144 & 85\end{bmatrix} - \begin{bmatrix}60 & 36 \\ 144 & 84\end{bmatrix} - \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow A^2 - 12A - I = \begin{bmatrix}61 - 60 - 1 & 36 - 36 + 0 \\ 144 - 144 + 0 & 85 - 84 - 1\end{bmatrix}\]
\[ \Rightarrow A^2 - 12A - I = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
Since A is satisfying the equation `A^2 - 12A - I,` A is the root of the equation `A^2 - 12A - I`