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If $A=\begin{bmatrix}1&0\\ 2&4\end {bmatrix}$ then find the eigen values of $4A^{-1}+ 3A + 21$
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Since A is a lower triangular Matrix, Eigen values (λ)= diagonal elements

$\therefore λ=1,4$

We know ,$f(λ)$ is eigen value of $f(A)$

Let $f(A)=4A^{-1}+3A+2I\\ \therefore f(λ)=4λ^{-1}+3λ+2\\ =\dfrac 4λ+3λ+2$

When $λ=1\\ f(1)4+3+2=7+2=9$

When $λ=4\\ f(4)=\dfrac 44+3(4)+2=1+12+2=15$

$\therefore $ Eigen values of $4A^{-1}+3A+2I \space \space are \space \space 9,15$

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