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Let, A = [ i −i ; −1 1 ], i = √−1. Then the system of linear equations A^8 [ x ; y ] = [ 8 ; 64 ] has -

1. A unique solution
2. Infinitely many solutions
3. No solution
4. Exactly two solutions

Correct answer: Exactly two solutions

Solution

The matrix A is a 2x2 matrix with a determinant that is non-zero, indicating it is invertible. When raised to the power of 8, it remains invertible, and since the right-hand side vector [8; 64] does not lie in the column space of A^8, the system has exactly two solutions due to the nature of complex eigenvalues.

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