If A = [ * 20 c a&c &b ],then A^ - 1 = - Vidyadip

MCQ

If $A = \left[ {\begin{array}{*{20}{c}}a&c\\d&b\end{array}} \right],$then ${A^{ - 1}}$=

✓
$\frac{1}{{ab - cd}}\left[ {\begin{array}{*{20}{c}}b&{ - c}\\{ - d}&a\end{array}} \right]$
B
$\frac{1}{{ad - bc}}\left[ {\begin{array}{*{20}{c}}b&{ - c}\\{ - d}&a\end{array}} \right]$
C
$\frac{1}{{ab - cd}}\left[ {\begin{array}{*{20}{c}}b&d\\c&a\end{array}} \right]$
D
None of these

✓

Answer

Correct option: A.

$\frac{1}{{ab - cd}}\left[ {\begin{array}{*{20}{c}}b&{ - c}\\{ - d}&a\end{array}} \right]$

a
(a) ${A^{ - 1}} = \frac{{adj\,A}}{{|A|}}$
But $|A| = \left| {\,\begin{array}{*{20}{c}}a&c\\d&b\end{array}\,} \right| = ab - cd$ and $adj\,A = \left[ {\begin{array}{*{20}{c}}b&{ - c}\\{ - d}&a\end{array}} \right]$
therefore ${A^{ - 1}} = \frac{1}{{ab - cd}}\left[ {\begin{array}{*{20}{c}}b&{ - c}\\{ - d}&a\end{array}} \right]$.

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