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  4. If A=[2 λ -3 0 2 5 1 1 3], the then A-1 exists, if

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Q. If $A=\begin{bmatrix}2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{bmatrix}$, the then $A^{-1}$ exists, if

Determinants

Solution:

The given matrix is $A=\begin{bmatrix}2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{bmatrix}$.
Now, $A^{-1}$ exists, if $|A| \neq 0$
$\Rightarrow \begin{vmatrix}2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{vmatrix} \neq 0$
$\Rightarrow 2\begin{vmatrix}2 & 5 \\ 1 & 3\end{vmatrix}-\lambda\begin{vmatrix}0 & 5 \\ 1 & 3\end{vmatrix}-3\begin{vmatrix}0 & 2 \\ 1 & 1\end{vmatrix}\neq 0$
$\Rightarrow 2(6-5)-\lambda(0-5)-3(0-2) \neq 0 $
$ \Rightarrow 2+5 \lambda+6 \neq 0 $
$ \Rightarrow 5 \lambda \neq-8 $
$ \Rightarrow \lambda \neq-\frac{8}{5}$