JEE MAIN - Mathematics (2017 (Offline) - No. 5)
If $$A = \left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$,
then adj(3A2 + 12A) is equal to
then adj(3A2 + 12A) is equal to
$$\left[ {\matrix{
{51} & {63} \cr
{84} & {72} \cr
} } \right]$$
$$\left[ {\matrix{
{51} & {84} \cr
{63} & {72} \cr
} } \right]$$
$$\left[ {\matrix{
{72} & {-63} \cr
{-84} & {51} \cr
} } \right]$$
$$\left[ {\matrix{
{72} & {-84} \cr
{-63} & {51} \cr
} } \right]$$
Explanation
We have, $$A = \left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$
$$ \therefore $$ A2 = A.A = $$\left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]\left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]$$
= $$\left[ {\matrix{ {4 + 12} & { - 6 - 3} \cr { - 8 - 4} & {12 + 1} \cr } } \right]$$
= $$\left[ {\matrix{ {16} & { - 9} \cr { - 12} & {13} \cr } } \right]$$
Now, 3A2 + 12A
= $$3\left[ {\matrix{ {16} & { - 9} \cr { - 12} & {13} \cr } } \right] + 12\left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]$$
= $$\left[ {\matrix{ {48} & { - 27} \cr { - 36} & {39} \cr } } \right] + \left[ {\matrix{ {24} & { - 36} \cr { - 48} & {12} \cr } } \right]$$
= $$\left[ {\matrix{ {72} & { - 63} \cr { - 84} & {51} \cr } } \right]$$
$$ \therefore $$ adj(3A2 + 12A) = $$\left[ {\matrix{ {51} & {63} \cr {84} & {72} \cr } } \right]$$
$$ \therefore $$ A2 = A.A = $$\left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]\left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]$$
= $$\left[ {\matrix{ {4 + 12} & { - 6 - 3} \cr { - 8 - 4} & {12 + 1} \cr } } \right]$$
= $$\left[ {\matrix{ {16} & { - 9} \cr { - 12} & {13} \cr } } \right]$$
Now, 3A2 + 12A
= $$3\left[ {\matrix{ {16} & { - 9} \cr { - 12} & {13} \cr } } \right] + 12\left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]$$
= $$\left[ {\matrix{ {48} & { - 27} \cr { - 36} & {39} \cr } } \right] + \left[ {\matrix{ {24} & { - 36} \cr { - 48} & {12} \cr } } \right]$$
= $$\left[ {\matrix{ {72} & { - 63} \cr { - 84} & {51} \cr } } \right]$$
$$ \therefore $$ adj(3A2 + 12A) = $$\left[ {\matrix{ {51} & {63} \cr {84} & {72} \cr } } \right]$$
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