Question

if A = $\begin{bmatrix}1&3\\ 3&4\end{bmatrix}$ 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 $

$\Rightarrow kA=\begin{bmatrix}1&3\\ 3&4\end{bmatrix}\begin{bmatrix}1&3\\ 3&4\end{bmatrix}-5\begin{bmatrix}1&0\\ 0&1\end{bmatrix}$

=$\begin{bmatrix}10&15\\ 15&25\end{bmatrix}-\begin{bmatrix}5&0\\ 0&5\end{bmatrix}=\begin{bmatrix}5&15\\ 15&20\end{bmatrix}=5\begin{bmatrix}1&3\\ 3&4\end{bmatrix}=5A$

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