Choose the correct answer from given four options in each of the … - Vidyadip

MCQ

Choose the correct answer from given four options in each of the Exercise: If $\text{A}=\begin{vmatrix}2&\lambda&-3\\0&2&5\\1&1&3\end{vmatrix},$ then $A^{-1}$ exists, if:

A
$\lambda=2$
B
$\lambda\neq2$
C
$\lambda\neq-2$
✓
$\text{None of these}$

✓

Answer

Correct option: D.

$\text{None of these}$

We have, $\text{A}=\begin{vmatrix}2&\lambda&-3\\0&2&5\\1&1&3\end{vmatrix}$
Expanding along $R_1,$ we get
$\text{A}=2(6-5)-\lambda(-5)-3(-2)$
$=2+5\lambda+6$
$=5\lambda+8$
We know that, $A^{-1}$ exists, if $A$ is non $-$ singular matrix
i.e., $|\text{A}|\neq0.$
$\therefore\ 5\lambda+8\neq0$
$\Rightarrow\ 5\lambda\neq-8$
$\therefore\ \lambda\neq\frac{-8}{5}$
Thus, $A^{-1}$ exists for all values of $\lambda\text{ except }\frac{-8}{5}.$

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