The element of second row and third column in the inverse of [ * … - Vidyadip

MCQ

The element of second row and third column in the inverse of $\left[ {\begin{array}{*{20}{c}}1&2&1\\2&1&0\\{ - 1}&0&1\end{array}} \right]$ is

A
$-2$
✓
$-1$
C
$1$
D
$2$

✓

Answer

Correct option: B.

$-1$

b
(b)In ${A^{ - 1}},$ the element of $2^{nd}$ row and $ 3^{rd}$ column is the ${c_{32}}$ element of the matrix $({c_{ij}})$ of cofactors of element of $Adj\,(A) = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&4\\4&1&{ - 2}\\{ - 2}&4&1\end{array}} \right]$, (due to transposition) divided by $\Delta = \,|A|\, = - 2$.
$\therefore $ Required element = $\frac{{{{( - 1)}^{3 + 2}}{M_{32}}}}{{ - 2}}\,{\rm{ }} = \frac{{ - ( - 2)}}{{ - 2}} = - 1$,
where ${M_{32}} = $minor of ${c_{32}}$ in $A = \left[ {\begin{array}{*{20}{c}}
1&1 \\
2&0
\end{array}} \right] = 0 - 2 = - 2$

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