Question

Let $A=\left[\begin{array}{cc}5& 8\\ 8& 13\end{array}\right]$ then show that $A$ satisfies the equation $x^2-18x+1=0$.

Answer 1

Given,

$A=\left[\begin{array}{cc}5& 8\\ 8& 13\end{array}\right]$.

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

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

$\left|\begin{array}{cc}5-a& 8\\ 8& 13-a\end{array}\right|=0$

or, $a^2-18a+1=0$.

According to the Caley-Hamilton's theorem we have " Every square matrix satisfies its own characteristic equation.".

Then $A$ will satisfy the equation $x^2-18x+1=0$.