If A = [ * 20 c - 1 &2 2& - 1 ] and B = [ l 3 1 ],AX = B, then X … - Vidyadip

MCQ

If $A = \left[ {\begin{array}{*{20}{c}}{ - 1}&2\\2&{ - 1}\end{array}} \right]$ and $B = \left[ \begin{array}{l}3\\1\end{array} \right],AX = B,$ then $X = $

A
$[5 7]$
✓
$\frac{1}{3}\left[ \begin{array}{l}5\\7\end{array} \right]$
C
$\frac{1}{3}[5\,\,7]$
D
$\left[ \begin{array}{l}5\\7\end{array} \right]$

✓

Answer

Correct option: B.

$\frac{1}{3}\left[ \begin{array}{l}5\\7\end{array} \right]$

$A = \left[ {\,\begin{array}{*{20}{c}}{ - 1}&2\\2&{ - 1}\end{array}\,} \right]\,\,,\,\,\,B = \left[ \begin{array}{l}3\\1\end{array} \right]$
$AX = B\,$
$\Rightarrow X = {A^{ - 1}}B$
${A^{ - 1}} = \frac{{\text{adj}\,A}}{{|A|}}$
${A^{ - 1}} = \frac{{ - 1}}{3}\left[ {\,\begin{array}{*{20}{c}}{ - 1}&{ - 2}\\{ - 2}&{ - 1}\end{array}\,} \right]$$ = \frac{1}{3}\left[ {\,\begin{array}{*{20}{c}}1&2\\2&1\end{array}\,} \right]$
and $X = {A^{ - 1}}B$ = $\left[ {\,\begin{array}{*{20}{c}}{\frac{1}{3}}&{\frac{2}{3}}\\{\frac{2}{3}}&{\frac{1}{3}}\end{array}\,} \right]\,\left[ \begin{array}{l}3\\1\end{array} \right]$
$X = \frac{1}{3}\left[ \begin{array}{l}5\\7\end{array} \right]$.

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