If `A= [[0, 1],[-1, 0]],` find `x`and `y` such that `(x I+y A)^2=A`

Explanation

To solve the problem, we need to find the values of \( x \) and \( y \) such that \( (xI + yA)^2 = A \), where \( A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \) and \( I \) is the identity matrix.

### Step-by-Step Solution:

1. **Define the matrices**:
\[ I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]
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