Question

Let $f\left(x\right)=x^2-5x+5$ then find $f\left(A\right)$ for $A=\left[\begin{array}{cc}3& 1\\ 1& 2\end{array}\right]$.

Answer 1

Given the matrix, $A=\left[\begin{array}{cc}3& 1\\ 1& 2\end{array}\right]$.

Now the characteristic equation corresponding to the matrix $A$ is,

$|A-aI|=0$ [ Where $I$ is the unit matrix of order $2$ and $a$ is a scalar]

or, $\left|\begin{array}{cc}3-a& 1\\ 1& 2-a\end{array}\right|=0$

or, $(3-a)(2-a)-1=0$

or, $a^2-5a+5=0$

Now by Caley Hamilton's theorem every square matrix satisfies its own characteristic equation.

Then $A^2-5A+5I=O$. [ Where $O$ is the zero matrix of order $2$]

or, $f\left(A\right)=O$.