Question

If $A=\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right],f\left(x\right)=x^2-5x+4$ then find value of $|f\left(A\right)|$

Answer 1

$A=\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]f\left(x\right)=x^2-5x+4$

$f\left(A\right)={\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]}^2-5\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]+4\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$f\left(A\right)=\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}10& 0\\ 0& 5\end{array}\right]+\left[\begin{array}{cc}4& 0\\ 0& 4\end{array}\right]$

$f\left(A\right)=\left[\begin{array}{cc}2\times 2+0& 2\times 0+0\times 1\\ 0\times 2+1\times 0& 0\times 0+1\times 1\end{array}\right]-\left[\begin{array}{cc}10& 0\\ 0& 5\end{array}\right]+\left[\begin{array}{cc}4& 0\\ 0& 4\end{array}\right]$

$f\left(A\right)=\left[\begin{array}{cc}4& 0\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}10& 0\\ 0& 5\end{array}\right]+\left[\begin{array}{cc}4& 0\\ 0& 4\end{array}\right]$

$f\left(A\right)=\left[\begin{array}{cc}4-10+4& 0-0+0\\ 0-0+0& 1-5+4\end{array}\right]$

$f\left(A\right)=\left[\begin{array}{cc}-2& 0\\ 0& 0\end{array}\right]$

$\therefore |f\left(A\right)|=0$