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DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS1 Mark
MCQ
If $\text{A}=\begin{bmatrix}2&\lambda&-3\\0&2&5\\1&1&3\end{bmatrix},$ then $\text{A}^{-1}$ exists if:
  • A
    $\lambda=2$
  • B
    $\lambda\neq2$
  • C
    $\lambda\neq-2$
  • ✓
    $\text{None of these}$
✓

Answer

Correct option: D.
$\text{None of these}$
$\text{A}=\begin{bmatrix}2&\lambda&-3\\0&2&5\\1&1&3\end{bmatrix}$

The inverse of a matrix exists if its determinant is not equal to 0.

Consider,

$|\text{A}|=\begin{bmatrix}2&\lambda&-3\\0&2&5\\1&1&3\end{bmatrix}\neq0$

$\Rightarrow|\text{A}| = 2 (6 – 5) – \lambda (0 – 5) + (-3) (0 – 2)\neq0$

$\Rightarrow2 + 5\lambda + 6 \neq 0$

$\Rightarrow5\lambda + 8 \neq 0$

$\Rightarrow5\lambda \neq -8$

$\Rightarrow\lambda\neq\frac{-8}{5}$

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