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Question

Mathematics Question on Matrices

if A = [13 34]\begin{bmatrix}1&3\\\ 3&4\end{bmatrix} and A2−kA−5I=0A^2 - kA - 5I = 0, then kk =

A

55

B

33

C

77

D

None of these

Answer

55

Explanation

Solution

Given A2−kA−5I=0A^{2} -kA- 5I= 0 ⇒kA=A2−5I\Rightarrow kA=A^{ 2 }-5I ⇒kA=[13 34][13 34]−5[10 01]\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} =[1015 1525]−[50 05]=[515 1520]=5[13 34]=5A\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 ⇒kA=5A∴k=5\Rightarrow kA = 5A \therefore k=5