Matriisirasaiteet

Matriisirasaiteet, known in English as matrix multiplication, is a fundamental operation in linear algebra. It involves multiplying two matrices, say matrix A and matrix B, to produce a third matrix, C. The dimensions of the matrices are crucial for this operation to be defined. If matrix A has dimensions m x n (m rows and n columns), then matrix B must have dimensions n x p (n rows and p columns). The resulting matrix C will have dimensions m x p.

The element at row i and column j of the resulting matrix C, denoted as c_ij, is calculated by taking the dot product of the i-th row of matrix A and the j-th column of matrix B. This means you multiply each element in the i-th row of A by the corresponding element in the j-th column of B and then sum up these products. For example, if the i-th row of A is [a_i1, a_i2, ..., a_in] and the j-th column of B is [b_1j, b_2j, ..., b_nj], then c_ij = (a_i1 * b_1j) + (a_i2 * b_2j) + ... + (a_in * b_nj).

Matrix multiplication is not commutative, meaning that in general, A * B is not equal to B * A. However, it is associative, so (A * B) * C = A * (B * C). This operation has widespread applications in various fields, including computer graphics, data analysis, physics, and engineering, where it is used to represent transformations, solve systems of linear equations, and model complex systems.