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Question
\[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix} and \text{ I} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
Solution
Given : `A =[[3 1],[-1 2]]`
Now,
`A^2`=AA
`⇒A^2= [[3 1],[-1 2]] [[3 1],[-1 2]]`
`⇒A^2=[[9-1 3+2],[-3-2 -1+4]]`
`⇒A^2=[[8 5],[-5 3]]`
`A^2=5A+λI`
⇒`[[8 5],[-5 3]]=5[[3 1],[-1 2]]+λ[[1 0],[0 1]]`
⇒`[[8 5],[-5 3]]` =`[[15 5],[-5 10]]+[[λ 0],[0 λ]]`
⇒`[[8 5],[-5 3]]`=`[[15+λ 5+0],[-5+0 10+λ]]`
⇒`[[8 5],[-5 3]]``[[15+λ 5+0],[-5+0 10+λ]]`
The corresponding elements of two equal matrices are equal.
∴ 8=15+λ
⇒8−15=λ
⇒−7=λ
∴ λ=−7
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