JEE MAIN - Mathematics (2016 - 10th April Morning Slot - No. 4)
If A = $$\left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]$$,
then the determinant of the matrix (A2016 − 2A2015 − A2014) is :
then the determinant of the matrix (A2016 − 2A2015 − A2014) is :
2014
$$-$$ 175
2016
$$-$$ 25
Explanation
Given,
$$A = \left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$
$${A^2} = \left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]\left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$
$$ = \left[ {\matrix{ {13} & 3 \cr { - 9} & { - 2} \cr } } \right]$$
A2 $$-$$ 2A $$-$$ I
$$ = \left[ {\matrix{ {13} & 3 \cr { - 9} & { - 2} \cr } } \right] - \left[ {\matrix{ { - 8} & { - 2} \cr 6 & 2 \cr } } \right] - \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right]$$
$$ = \left[ {\matrix{ {20} & 5 \cr { - 15} & { - 5} \cr } } \right]$$
$$\left| A \right| = \left| {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right|$$ $$=$$ $$-$$ 4 + 3 $$=$$ $$-$$ 1
Now,
$$\left| {{A^{2016}} - 2{A^{2015}} - {A^{2014}}} \right|$$
$$=$$ $${\left| A \right|^{2014}}\left| {{A^2} - 2A - {\rm I}} \right|$$
$$ = {\left( { - 1} \right)^{2014}}\left| {\matrix{ {20} & 5 \cr { - 15} & { - 5} \cr } } \right|$$
$$=$$ 1 $$ \times $$ ($$-$$ 100 + 75)
$$=$$ $$-$$ 25
$$A = \left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$
$${A^2} = \left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]\left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$
$$ = \left[ {\matrix{ {13} & 3 \cr { - 9} & { - 2} \cr } } \right]$$
A2 $$-$$ 2A $$-$$ I
$$ = \left[ {\matrix{ {13} & 3 \cr { - 9} & { - 2} \cr } } \right] - \left[ {\matrix{ { - 8} & { - 2} \cr 6 & 2 \cr } } \right] - \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right]$$
$$ = \left[ {\matrix{ {20} & 5 \cr { - 15} & { - 5} \cr } } \right]$$
$$\left| A \right| = \left| {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right|$$ $$=$$ $$-$$ 4 + 3 $$=$$ $$-$$ 1
Now,
$$\left| {{A^{2016}} - 2{A^{2015}} - {A^{2014}}} \right|$$
$$=$$ $${\left| A \right|^{2014}}\left| {{A^2} - 2A - {\rm I}} \right|$$
$$ = {\left( { - 1} \right)^{2014}}\left| {\matrix{ {20} & 5 \cr { - 15} & { - 5} \cr } } \right|$$
$$=$$ 1 $$ \times $$ ($$-$$ 100 + 75)
$$=$$ $$-$$ 25
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