Question

If $A=\left[\begin{array}{cc}2& -1\\ 1& 3\end{array}\right]$, evaluating $A^2-3A+21$ where $I$ is a unit matrix of order $2.\left(4\right)$.

Answer 1

$\begin{array}{c}{A}^2=A.A=\left[\begin{array}{c}2-1\\ 13\end{array}\right]\left[\begin{array}{c}2-1\\ 13\end{array}\right]\\ \Rightarrow \left[\begin{array}{c}4-1-2-3\\ 2+3-1+9\end{array}\right]=\left[\begin{array}{c}3-5\\ 58\end{array}\right]\\ 3A\Rightarrow 3\left[\begin{array}{c}2-1\\ 13\end{array}\right]\Rightarrow \left[\begin{array}{c}6-3\\ 39\end{array}\right]\\ 2I=2\left[\begin{array}{c}10\\ 01\end{array}\right]\Rightarrow \left[\begin{array}{c}20\\ 02\end{array}\right]\\ puttingallthevalueinabove,\\ weget\\ \Rightarrow \left[\begin{array}{c}3-5\\ 58\end{array}\right]-\left[\begin{array}{c}6-3\\ 02\end{array}\right]+\left[\begin{array}{c}20\\ 02\end{array}\right]\\ \Rightarrow \left[\begin{array}{c}-3-2\\ 2-1\end{array}\right]+\left[\begin{array}{c}20\\ 02\end{array}\right]\\ \Rightarrow \left[\begin{array}{c}-1-2\\ 21\end{array}\right]Ans.\end{array}$