Let A = [ * 20 c x + &x&x & x + &x &x& x + ] , then A^ -1 exists … - Vidyadip

MCQ

Let $A =$ $\left[ {\begin{array}{*{20}{c}}{x + \lambda }&x&x\\x&{x + \lambda}&x\\x&x&{x + \lambda }\end{array}} \right]$ , then $A^{-1}$ exists if

A
$x \ne 0$
B
$\lambda \ne 0$
✓
$3x + \lambda \ne 0, \lambda \ne 0$
D
$x \ne 0, \lambda \ne 0$

✓

Answer

Correct option: C.

$3x + \lambda \ne 0, \lambda \ne 0$

c
We have $|A| =$ $\left[ {\begin{array}{*{20}{c}}{x + \lambda }&x&x\\x&{x +\lambda }&x\\x&x&{x + \lambda }\end{array}} \right]$ $=$ $\left[{\begin{array}{*{20}{c}}{3x + \lambda }&x&x\\{3x + \lambda }&{x + \lambda }&x\\{3x + \lambda }&x&{x + \lambda }\end{array}} \right]$ $=$ $(3x +\lambda )$ $\left[ {\begin{array}{*{20}{c}}1&x&x\\1&{x + \lambda }&x\\ 1&x&{x + \lambda }\end{array}} \right]$

$=$ $(3x + \lambda )$ $\left[{\begin{array}{*{20}{c}}1&x&x\\0&\lambda &0\\0&0&\lambda \end{array}} \right]$ $= \lambda ^2(3x + \lambda )$ [Take $3x + \lambda$ common and use $R_2 \rightarrow R_2 -R_1, R_3 \rightarrow R_3- R_1$]

Thus, $A^{-1}$ will exist if $\lambda \ne 0$ and $3x + \lambda \ne 0$

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