"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 ____________."

Question

Accepted Answer

"$$P = \left[ {\matrix{ 2 & { - 1} \cr 5 & { - 3} \cr } } \right]$$

$$\left| {\matrix{ {2 - \lambda } & { - 1} \cr 5 & { - 3 - \lambda } \cr } } \right| = 0$$

$\Rightarrow$ $\lambda$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"

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 ____________.

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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 ____________.

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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 ____________.