If A= [ ccc 2 & & -3 0 & 2 & 5 1 & 1 & 3 ], then A^ -1 exists onl… - Vidyadip

MCQ

If $A=\left[\begin{array}{ccc}2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{array}\right]$, then $A^{-1}$ exists only if

A
$\lambda=2$
B
$\lambda \neq 2$
C
$\lambda \neq-2$
✓
$\lambda \neq-8 / 5$

✓

Answer

Correct option: D.

$\lambda \neq-8 / 5$

$A^{-1}$ exists if $|A| \neq 0$
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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