The matrix [ * 20 c 2& & - 4 - 1 &3&4 1& - 2 & - 3 ]is non singul… - Vidyadip

MCQ

The matrix $\left[ {\begin{array}{*{20}{c}}2&\lambda &{ - 4}\\{ - 1}&3&4\\1&{ - 2}&{ - 3}\end{array}} \right]$is non singular, if

✓
$\lambda \ne - 2$
B
$\lambda \ne 2$
C
$\lambda \ne 3$
D
$\lambda \ne - 3$

✓

Answer

Correct option: A.

$\lambda \ne - 2$

a
(a) The given matrix $A = \left[ {\begin{array}{*{20}{c}}2&\lambda &{ - 4}\\{ - 1}&3&4\\1&{ - 2}&{ - 3}\end{array}} \right]$ is non singular, if $|A|\, \ne 0$

$|A|\,\, = \left| {\,\begin{array}{*{20}{c}}2&\lambda &{ - 4}\\{ - 1}&3&4\\1&{ - 2}&{ - 3}\end{array}\,} \right|\,$=$\left| {\,\begin{array}{*{20}{c}}1&{\lambda + 3}&0\\{ - 1}&3&4\\1&{ - 2}&{ - 3}\end{array}\,} \right|\,$,$[{R_1} \to {R_2} + {R_1}]$

= $\left| {\,\begin{array}{*{20}{c}}1&{\lambda + 3}&0\\0&1&1\\0&{ - \lambda - 5}&{ - 3}\end{array}\,} \right|$$\left[ {\begin{array}{*{20}{c}}{{R_2} \to {R_2} + {R_3}}\\{{R_3} \to {R_3} - {R_1}}\end{array}} \right]$

= $1\,( - 3 + \lambda + 5) \ne 0$

$ \Rightarrow \lambda + 2 \ne 0$ $ \Rightarrow \lambda \,\, \ne - 2.$

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