Matrix Chain Multiplication is a classical problem in computer science and optimization that involves finding the most efficient way to multiply a given sequence of matrices. The goal is to minimize the total number of scalar multiplications needed to compute the product of the matrices.
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Given a fixed sequence of compatible matrices , choose a binary parenthesization of that minimizes the number of scalar multiplications in ordinary matrix multiplication. The order of the matrices is fixed: parenthesization changes the grouping, not the operand order. Associativity preserves the product while intermediate matrix dimensions change the cost. The possible binary groupings are counted by the Catalan number for .