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The algebra of matrices.

As we saw previously, a matrix is a rectangular array of elements of ; we say that the matrix is of order if it has m rows and n columns. The set of all matrices of order with entries in is denoted .

The matrix is denoted , or (aij) if no confusion is possible.

If m=n, the matrix is a square matrix of order n.

Equality:

Let A=(aij) and B=(bij) be two matrices in .

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Addition:

Let A=(aij), B=(bij) and C=(cij) be three matrices in .

The zero matrix: the zero matrix in is the matrix (aij)such that . Multiplication of a matrix by a scalar: Let A=(aij) and B=(bij) be two matrices in and take .

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Multiplication:

Let , and .

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This definition is motivated by the link between matrices and linear operators: the product of two matrices corresponds to the composition of two linear operators (v.s.  1.4).

Next: How to compute the Up: No Title Previous: Linear transformations and matrices.
Noah Dana-Picard
2001-02-26