Question

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

• A. $5$
• B. $3$
• C. $7$
• D. None of these

Answer 1

Answer: (A)

Given $A^2-kA-5I=0$

$\Rightarrow kA=A^2-5I$

$\Rightarrow kA=\left[\begin{array}{cc}1& 3\\ 3& 4\end{array}\right]\left[\begin{array}{cc}1& 3\\ 3& 4\end{array}\right]-5\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

=$\left[\begin{array}{cc}10& 15\\ 15& 25\end{array}\right]-\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]=\left[\begin{array}{cc}5& 15\\ 15& 20\end{array}\right]=5\left[\begin{array}{cc}1& 3\\ 3& 4\end{array}\right]=5A$

$\Rightarrow kA=5A\therefore k=5$