JEE MAIN - Mathematics (2019 - 12th April Morning Slot - No. 15)
If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = $$\left[ {\matrix{
2 & 3 \cr
5 & { - 1} \cr
} } \right]$$, then AB is equal
to :
$$\left[ {\matrix{
4 & { - 2} \cr
1 & { - 4} \cr
} } \right]$$
$$\left[ {\matrix{
{ - 4} & { - 2} \cr
{ - 1} & 4 \cr
} } \right]$$
$$\left[ {\matrix{
{ - 4} & 2 \cr
1 & 4 \cr
} } \right]$$
$$\left[ {\matrix{
4 & { - 2} \cr
{ - 1} & { - 4} \cr
} } \right]$$
Explanation
$$A + B = \left[ {\matrix{
2 & 3 \cr
5 & { - 1} \cr
} } \right] = P(say)$$
Now $$A = {{P + {P^T}} \over 2}\& B = {{P - {P^T}} \over 2}$$
So $$A = {1 \over 2}\left( {\left[ {\matrix{ 2 & 3 \cr 5 & { - 1} \cr } } \right] + \left[ {\matrix{ 2 & 5 \cr 3 & { - 1} \cr } } \right]} \right) = \left[ {\matrix{ 2 & 4 \cr 4 & { - 1} \cr } } \right]$$
$$B = {1 \over 2}\left( {\left[ {\matrix{ 2 & 3 \cr 5 & { - 1} \cr } } \right] - \left[ {\matrix{ 2 & 5 \cr 3 & { - 1} \cr } } \right]} \right) = \left[ {\matrix{ 0 & { - 1} \cr 1 & 0 \cr } } \right]$$
So $$AB = \left( {\left[ {\matrix{ 2 & 4 \cr 4 & { - 1} \cr } } \right]\left[ {\matrix{ 0 & { - 1} \cr 1 & 0 \cr } } \right]} \right) = \left[ {\matrix{ 4 & { - 2} \cr { - 1} & { - 4} \cr } } \right]$$
Now $$A = {{P + {P^T}} \over 2}\& B = {{P - {P^T}} \over 2}$$
So $$A = {1 \over 2}\left( {\left[ {\matrix{ 2 & 3 \cr 5 & { - 1} \cr } } \right] + \left[ {\matrix{ 2 & 5 \cr 3 & { - 1} \cr } } \right]} \right) = \left[ {\matrix{ 2 & 4 \cr 4 & { - 1} \cr } } \right]$$
$$B = {1 \over 2}\left( {\left[ {\matrix{ 2 & 3 \cr 5 & { - 1} \cr } } \right] - \left[ {\matrix{ 2 & 5 \cr 3 & { - 1} \cr } } \right]} \right) = \left[ {\matrix{ 0 & { - 1} \cr 1 & 0 \cr } } \right]$$
So $$AB = \left( {\left[ {\matrix{ 2 & 4 \cr 4 & { - 1} \cr } } \right]\left[ {\matrix{ 0 & { - 1} \cr 1 & 0 \cr } } \right]} \right) = \left[ {\matrix{ 4 & { - 2} \cr { - 1} & { - 4} \cr } } \right]$$
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