Question

If $A=\left[\begin{array}{cc}1& 3\\ 3& 4\end{array}\right]$ and $A^2-kA-5I_2=0$ then $k=$

• A. $3$
• B. $5$
• C. $-5$
• D. $-3$

Answer 1

The correct option is B $5$

Given,

$A=\left[\begin{array}{cc}1& 3\\ 3& 4\end{array}\right]$

Now the characteristic equation of $A$ is

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

or, $(1-a)(4-a)-9=0$

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

By Cayley Hamilton's theorem we've every square matrix satisfies its own characteristic equation.

Then we get,

$A^2-5A-5I_2=0$.

Comparing this with the given equation we get, $k=5$.