Let \(\mathrm{A}=\left[\begin{array}{cc}\mathrm{i} & -\mathrm{i} \\ -\mathrm{i} & \mathrm{i}\end{array}\right], \mathrm{i}=\sqrt{-1}\). Then, the system of linear equation \(\mathrm{A}^{8}\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y}\end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]\) has :

JEE Main · 2021 · 16 Mar 2021 · Shift 1 · Matrices

Let

A = [i - i - i i], i = - 1

. Then, the system of linear equation

A^{8} [x y] = [864]

has :

Answer: (B)

Squaring the matrix,

A^{2} = [i - i - i i] [i - i - i i] = [- 2 2 2 - 2] = - 2 M, M = [1 - 1 - 1 1]

Since

M^{2} = 2 M

, it follows that

A^{4} = 4 M^{2} = 8 M

and

A^{8} = 64 M^{2} = 128 M

, so

A^{8} = [128 - 128 - 128 128]

The two equations are

128 (x - y) = 8

and

- 128 (x - y) = 64

, which require

x - y = \frac{1}{16}

and

x - y = - \frac{1}{2}

at once. These are contradictory, so the system has no solution.

JEE Main · 2021 · 16 Mar 2021 · Shift 1 · Single correct option