JEE MAIN - Mathematics (2021 - 18th March Evening Shift - No. 19)
Let I be an identity matrix of order 2 $$\times$$ 2 and P = $$\left[ {\matrix{
2 & { - 1} \cr
5 & { - 3} \cr
} } \right]$$. Then the value of n$$\in$$N for which Pn = 5I $$-$$ 8P is equal to ____________.
Answer
6
Explanation
$$P = \left[ {\matrix{
2 & { - 1} \cr
5 & { - 3} \cr
} } \right]$$
$$\left| {\matrix{ {2 - \lambda } & { - 1} \cr 5 & { - 3 - \lambda } \cr } } \right| = 0$$
$$ \Rightarrow $$ $$\lambda$$2 + $$\lambda$$ $$-$$ 1 = 0
$$ \Rightarrow $$ P2 + P $$-$$ I = 0
$$ \Rightarrow $$ P2 = I $$-$$ P
$$ \Rightarrow $$ P4 = I + P2 $$-$$ 2P
$$ \Rightarrow $$ P4 = 2I $$-$$ 3P
Now, P4 . P2 = (2I $$-$$ 3P)(I $$-$$ P) = 2I $$-$$ 5P + 3P2
$$ \Rightarrow $$ P6 = 5I $$-$$ 8P
So n = 6
$$\left| {\matrix{ {2 - \lambda } & { - 1} \cr 5 & { - 3 - \lambda } \cr } } \right| = 0$$
$$ \Rightarrow $$ $$\lambda$$2 + $$\lambda$$ $$-$$ 1 = 0
$$ \Rightarrow $$ P2 + P $$-$$ I = 0
$$ \Rightarrow $$ P2 = I $$-$$ P
$$ \Rightarrow $$ P4 = I + P2 $$-$$ 2P
$$ \Rightarrow $$ P4 = 2I $$-$$ 3P
Now, P4 . P2 = (2I $$-$$ 3P)(I $$-$$ P) = 2I $$-$$ 5P + 3P2
$$ \Rightarrow $$ P6 = 5I $$-$$ 8P
So n = 6
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