Let \(\mathrm{A}=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]\). If M and N are two matrices given by \(M=\sum_{k=1}^{10} A^{2 k}\) and \(N=\sum_{k=1}^{10} A^{2 k-1}\) then \(\mathrm{MN}^{2}\) is

(A) a non-identity symmetric matrix

(C) neither symmetric nor skew-symmetric matrix

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Matrices

Let

A = [02 - 2 0]

. If M and N are two matrices given by

M = \sum_{k = 1}^{10} A^{2 k}

and

N = \sum_{k = 1}^{10} A^{2 k - 1}

then

MN^{2}

Answer: (A)

Computing

A^{2} = [02 - 2 0] [02 - 2 0] = [- 4 0 0 - 4] = - 4 I

. Thus

A^{2 k} = (- 4)^{k} I

and

A^{2 k - 1} = (- 4)^{k - 1} A

. So

M = \sum_{k = 1}^{10} (- 4)^{k} I = - 4 λ I

(symmetric), and

N = \sum_{k = 1}^{10} (- 4)^{k - 1} A = μ A

(skew-symmetric). Then

N^{2} = μ^{2} A^{2} = - 4 μ^{2} I

(symmetric). Therefore

M N^{2} = (- 4 λ I) (- 4 μ^{2} I) = 16 λ μ^{2} I

, which is a non-identity symmetric matrix (since the scalar is non-zero and not 1).

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Single correct option