By Olver P.J.

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**Example text**

For example the product s = r c of a row vector r, a 1 × n matrix, and a column vector c, an n × 1 matrix with the same number of entries, is a 1 × 1 matrix or scalar, whereas the reversed product C = c r is an n × n matrix. For instance, (1 2) 3 0 = 3, 3 (1 2) = 0 whereas 3 0 6 0 . In computing the latter product, don’t forget that we multiply the rows of the first matrix by the columns of the second. Moreover, even if the matrix products A B and B A have the same size, which requires both A and B to be square matrices, we may still have A B = B A.

Proof : The matrix inverse equations A−1 A = I = A A−1 are sufficient to prove that A is the inverse of A−1 . D. 3/15/06 41 c 2006 Peter J. 6. If A and B are invertible matrices of the same size, then their product, A B, is invertible, and (A B)−1 = B −1 A−1 . 10) Note that the order of the factors is reversed under inversion. Proof : Let X = B −1 A−1 . Then, by associativity, X (A B) = B −1 A−1 A B = B −1 B = I , (A B) X = A B B −1 A−1 = A A−1 = I . Thus X is both a left and a right inverse for the product matrix A B and the result follows.

A has n nonzero pivots. A admits a permuted L U factorization: P A = L U . A practical method to construct a permuted L U factorization of a given matrix A would proceed as follows. First set up P = L = I as n × n identity matrices. The matrix P will keep track of the permutations performed during the Gaussian Elimination process, while the entries of L below the diagonal are gradually replaced by the negatives of the multiples used in the corresponding row operations of type #1. Each time two rows of A are interchanged, the same two rows of P will be interchanged.