Let I be an identity matrix of order 2 × 2 and P = [ cases 2 & - 1 5 & - 3 cases ] . Then the value of n ∈ N for which P ^ n = 5I - 8P is equal to ___ .

Correct answer: 0 P = [ cases 2 & - 1 5 & - 3 cases ] 5I - 8P = [ cases 5 & 0 0 & 5 cases ] - [ cases 16 & - 8 40 & - 24 cases ] = [ cases - 11 & 8 - 40 & 29 cases ] P ^ 2 = [ cases 2 & - 1 5 & - 3 cases ][ cases 2 & - 1 5 & - 3 cases ] = [ cases - 1 & 1 - 5 & 4 cases ] P ^ 3 = [ cases 2 & - 1 5 & - 3 cases ][ cases - 1 & 1 - 5 & 4 cases ] = [ cases 3 & - 2 10 & - 7 cases ] ⇒ P ^ 6 = [ cases 3 & - 2 10 & - 7 cases ][ cases 3 & - 2 10 & - 7 cases ] = [ cases - 11 & 8 - 40 & 29 cases ] = P ^ n ⇒ n = 6

JEE Main 202118 MarchEvening Shift

Exam: JEE Main
Year: 2021
Date: 18 March
Shift: Evening Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2021 · 18 March · Evening Shift · Mathematics · Matrices

Let I be an identity matrix of order 2 × 2 and P = [ cases 2 & - 1...

Let

I

be an identity matrix of order

2 \times 2

and

P = \left[

\begin{cases}2 & - 1 \\ 5 & - 3\end{cases}

\right]

. Then the value of

n \in N

for which

{P}^{n} = 5I - 8P

is equal to ___ .

Correct answer

Step-by-step solution

$P = \left[$ \begin{cases}2 & - 1 \\ 5 & - 3\end{cases} $\right]$

$5I - 8P = \left[$ \begin{cases}5 & 0 \\ 0 & 5\end{cases} $\right] - \left[$ \begin{cases}16 & - 8 \\ 40 & - 24\end{cases} $\right] = \left[$ \begin{cases}- 11 & 8 \\ - 40 & 29\end{cases} $\right]$

${P}^{2} = \left[$ \begin{cases}2 & - 1 \\ 5 & - 3\end{cases} $\right]\left[$ \begin{cases}2 & - 1 \\ 5 & - 3\end{cases} $\right] = \left[$ \begin{cases}- 1 & 1 \\ - 5 & 4\end{cases} $\right]$

${P}^{3} = \left[$ \begin{cases}2 & - 1 \\ 5 & - 3\end{cases} $\right]\left[$ \begin{cases}- 1 & 1 \\ - 5 & 4\end{cases} $\right] = \left[$ \begin{cases}3 & - 2 \\ 10 & - 7\end{cases} $\right]$

$⇒ {P}^{6} = \left[$ \begin{cases}3 & - 2 \\ 10 & - 7\end{cases} $\right]\left[$ \begin{cases}3 & - 2 \\ 10 & - 7\end{cases} $\right] = \left[$ \begin{cases}- 11 & 8 \\ - 40 & 29\end{cases} $\right] = {P}^{n}$

$⇒ n = 6$

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