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

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

Q52 / 90 · JEE Main 2021 (18 Mar Shift 2) · Mathematics

Let $I$ be an identity matrix of order $2 \times 2$ and $P = [\begin{matrix} 2 & - 1 \\ 5 & - 3 \end{matrix}]$ . Then the value of $n \in N$ for which $P^{n} = 5 I - 8 P$ is equal to ___ .

Correct answer: 6

Solution

$P = [\begin{array}{l} 2 & - 1 \\ 5 & - 3 \end{array}]$

$5 I - 8 P = [\begin{array}{l} 5 & 0 \\ 0 & 5 \end{array}] - [\begin{matrix} 16 & - 8 \\ 40 & - 24 \end{matrix}] = [\begin{matrix} - 11 & 8 \\ - 40 & 29 \end{matrix}]$

$P^{2} = [\begin{array}{l} 2 & - 1 \\ 5 & - 3 \end{array}] [\begin{array}{l} 2 & - 1 \\ 5 & - 3 \end{array}] = [\begin{array}{l} - 1 & 1 \\ - 5 & 4 \end{array}]$

$P^{3} = [\begin{array}{l} 2 & - 1 \\ 5 & - 3 \end{array}] [\begin{array}{l} - 1 & 1 \\ - 5 & 4 \end{array}] = [\begin{matrix} 3 & - 2 \\ 10 & - 7 \end{matrix}]$

$\Rightarrow P^{6} = [\begin{matrix} 3 & - 2 \\ 10 & - 7 \end{matrix}] [\begin{matrix} 3 & - 2 \\ 10 & - 7 \end{matrix}] = [\begin{matrix} - 11 & 8 \\ - 40 & 29 \end{matrix}] = P^{n}$

$\Rightarrow n = 6$

Hence, the correct answer is 6.

Let I be an identity matrix of order 2×2 and P=2-15-3. Then the value of n∈N for which Pn=5I-8P is equal to ___ .

Solution

Relevant Questions

Let $I$ be an identity matrix of order $2 \times 2$ and $P = [\begin{matrix} 2 & - 1 \\ 5 & - 3 \end{matrix}]$ . Then the value of $n \in N$ for which $P^{n} = 5 I - 8 P$ is equal to ___ .