Question

If $A=\left[\begin{array}{cc}4& 1\\ -1& 2\end{array}\right]$ show that $6A-A^2=9I$

Answer 1

$\left[\begin{array}{cc}4& 1\\ -1& 2\end{array}\right]6A-A^2=9I$

$A^2=\left[\begin{array}{cc}4& 1\\ -1& 2\end{array}\right]\left[\begin{array}{cc}4& 1\\ -1& 2\end{array}\right]\Rightarrow \left[\begin{array}{cc}16-1& -2-4\\ 4+2& 1+4\end{array}\right]\left[\begin{array}{cc}16& -6\\ 6& 3\end{array}\right]$

$A^2=\left[\begin{array}{cc}4& 1\\ -1& 2\end{array}\right]\left[\begin{array}{cc}4& 1\\ -1& 2\end{array}\right]\Rightarrow \left[\begin{array}{cc}16-1& -2-4\\ 4+2& 1+4\end{array}\right]\left[\begin{array}{cc}16& -6\\ 6& 3\end{array}\right]$