Let A = [ i, - i; - i, i ],i = √(- 1) . Then, the system of linear equations A⁸[ x; y ] = [ 8; 64 ] has :

Solution

Maths sir · 4 steps

Pull i out of A first. The matrix left behind just doubles each time you square it.

1. Factor out i. Write A = iB with B = (1, -1; -1, 1). Then A⁸ = i⁸ B⁸ = B⁸, because i⁸ = (i⁴)² = 1.
2. Powers of B. Squaring gives B² = (2, -2; -2, 2) = 2B. Each extra power multiplies by 2 again, so Bⁿ = 2ⁿ⁻¹B.

   A⁸ = 2⁷ B = (128, -128; -128, 128)

3. Write the equations. The first row gives 128(x - y) = 8, so x - y = 1/16. The second row gives -128(x - y) = 64, so x - y = -1/2.
4. So. The same quantity x - y cannot equal both 1/16 and -1/2. The equations are inconsistent, so the system has no solution.

The idea to keep. When the determinant is zero, do not stop there. Write out the equations: they are either the same line (infinitely many) or contradict each other (no solution).

Answer from the NTA final answer key. Solution by PrepWiser.