Matrix Operations

A matrix is a rectangular array of numbers, and is called m by n when it has m rows and n columns. Examples include the matrices A (which is 2 by 3), M (3 by 2), and S (4 by 4) below:

Given an m by n matrix A, for each i = 1,…,m, and j = 1,…,n, the symbol aij denotes the entry of A in row i, column j.

For m by n matrices A and B, their sum, A + B, is the m by n matrix such that for all i = 1,…,m, j…, n, the entry in row i, column j is given by aij + bij . The sum of two matrices is defined only when both the number of rows and the number of columns of the summands agree. For example

For any m by n matrix A and any scalar x, the scalar multiple xA is the m by n matrix such that for all i= 1,…,m,j = 1,…,n, the entry in row i, column j is given by xaij . For example, if

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Addition and scalar multiplication can be combined for two m by n matrices A and B, so that the matrix xA + yB has the entries xaij + ybij for all i = 1,…,m,j = 1,…,n. Thus, if

For an m by n matrix U and an n by k matrix V, the matrix product UV is the m by k matrix such that for each i=1,…, m, j = 1,…,k, the entry in row i, column j of UV is given by Σnp=1uipvpj. The matrix product UV is defined only when the number of columns of U agrees with the number of rows of V . For example, if

If a matrix S has an equal number of rows and columns, we say that S is square. Given a square matrix S, for each natural number k, the kth power of S, denoted Sk, is the k-fold product of S with itself. For example, if

For each natural number n, the identity matrix of order n, denoted by In, is the n by n matrix with diagonal entries equal to 1 and all other entries equal to 0. For example,

For any m by n matrix A and any n by k matrix B, we have AIn = A and InB = B. Given an n by n matrix S, we say that S is invertible if there is an n by n matrix T such that ST = In and TS = In . If such a T exists, it is the inverse of S and is denoted by S-1. For any natural number k, the k-fold product of S-1 with itself is denotedS-k. For each natural number k, we have (Sk)-1 = S-k. For example if

Because of their computational utility and well-developed theory, matrices arise throughout the mathematical sciences and have numerous applications in science and engineering.

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