Matrix multiplication

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For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The result matrix has the number of rows of the first and the number of columns of the second matrix.

The values at the intersections, marked with circles in figure to the right, are:

Using same notation as above, such a system is equivalent with the single matrix equation

More generally, any bilinear form over a vector space of finite dimension may be expressed as a matrix product

These properties result from the bilinearity of the product of scalars:

where T denotes the transpose, that is the interchange of rows and columns.

This results from applying to the definition of matrix product the fact that the conjugate of a sum is the sum of the conjugates of the summands and the conjugate of a product is the product of the conjugates of the factors.

is defined and does not depend on the order of the multiplications, if the order of the matrices is kept fixed.

A product of matrices is invertible if and only if each factor is invertible. In this case, one has

One may raise a square matrix to any nonnegative integer power multiplying it by itself repeatedly in the same way as for ordinary numbers. That is,