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

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Matrices

Let I be an identity matrix of order

2 \times 2

and

P = [25 - 1 - 3]

. Then the value of

n \in N

for which

P^{n} = 5 I - 8 P

is equal to

Answer: 6

The matrix

P = [25 - 1 - 3]

has trace

- 1

and determinant

- 6 + 5 = - 1

, so its characteristic equation is

λ^{2} + λ - 1 = 0

and by the Cayley-Hamilton theorem

P^{2} + P - I = O ⟹ P^{2} = I - P

Successive powers follow by repeated substitution:

P^{3} = P \cdot P^{2} = P - P^{2} = 2 P - I

P^{4} = 2 P^{2} - P = 2 I - 3 P, P^{5} = 2 P - 3 P^{2} = 5 P - 3 I

P^{6} = 5 P^{2} - 3 P = 5 (I - P) - 3 P = 5 I - 8 P

so the required power is

n = 6

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Integer answer