If A is a symmetric matrix and B is a skew-symmetrix matrix such … - Vidyadip

MCQ

If $A$ is a symmetric matrix and $B$ is a skew-symmetrix matrix such that $A + B = \left[ {\begin{array}{*{20}{c}}
2&3\\
5&{ - 1}
\end{array}} \right]$ , then $AB$ is equal to

A
$\left[ {\begin{array}{*{20}{c}}
4&{ - 2}\\
1&{ - 4}
\end{array}} \right]$
✓
$\left[ {\begin{array}{*{20}{c}}
4&{ - 2}\\
{ - 1}&{ - 4}
\end{array}} \right]$
C
$\left[ {\begin{array}{*{20}{c}}
{ - 4}&2\\
1&4
\end{array}} \right]$
D
$\left[ {\begin{array}{*{20}{c}}
{ - 4}&{ - 2}\\
{ - 1}&4
\end{array}} \right]$

✓

Answer

Correct option: B.

$\left[ {\begin{array}{*{20}{c}}
4&{ - 2}\\
{ - 1}&{ - 4}
\end{array}} \right]$

b
$A = A',B = B'$

$A + B = \left[ {\begin{array}{*{20}{c}}
2&3\\
5&{ - 1}
\end{array}} \right]\,\,\,\,\,\,\,\,....\left( 1 \right)$

$A' + B' = \left[ {\begin{array}{*{20}{c}}
2&5\\
3&{ - 1}
\end{array}} \right]\,\,\,$

$A - B = \left[ {\begin{array}{*{20}{c}}
2&5\\
3&{ - 1}
\end{array}} \right]\,\,\,\,\,\,\,\,.....\left( 2 \right)$

After addding equation $(1)$ and $(2)$

$A = \left[ {\begin{array}{*{20}{c}}
2&4\\
4&{ - 1}
\end{array}} \right]\,,B = \left[ {\begin{array}{*{20}{c}}
0&{ - 1}\\
1&0
\end{array}} \right]\,$

$AB = \left[ {\begin{array}{*{20}{c}}
4&{ - 2}\\
{ - 1}&{ - 4}
\end{array}} \right]$

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