Question
If $A=\left[\begin{array}{ccc}2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{array}\right]$, then $A^{-1}$ exists only if
✓
Answer
$A^{-1}$ exists if $|A| \neq 0$
$\text { i.e., }\left|\begin{array}{ccc} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{array}\right| \neq 0$
$ \Rightarrow 2(6-5)+1(5 \lambda+6) \neq 0$
$\Rightarrow 2+5 \lambda+6 \neq 0 $
$\Rightarrow 5 \lambda \neq-8 $
$\text { i.e., } \lambda \neq \frac{-8}{5}$
$\text { i.e., }\left|\begin{array}{ccc} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{array}\right| \neq 0$
$ \Rightarrow 2(6-5)+1(5 \lambda+6) \neq 0$
$\Rightarrow 2+5 \lambda+6 \neq 0 $
$\Rightarrow 5 \lambda \neq-8 $
$\text { i.e., } \lambda \neq \frac{-8}{5}$
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