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Matrix multiplication is a fundamental operation in many algorithms, particularly in linear algebra and scientific computing. The conventional algorithm for multiplying two $n \times n$ matrices takes $O(n^3)$ time using a triple nested loop. However, Strassen's method significantly improves this by reducing the number of recursive multiplications required, at the cost of more additions and subtractions. Strassen's algorithm works by dividing each $n \times n$ matrix into four $n/2 \times n/2$ submatrices. Let $A$ and $B$ be the input matrices to be multiplied, and $C = A \times B$. We write: $$ A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}, \quad B = \begin{bmatrix} B_{11} & B_{12} ________ ____ _________ __________ ______ ______ ______ _________ ________.
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