College Algebra Matrices
Matrix Multiplication Concepts 1

|  |  | Statement |  |
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If A = (aij)m x n and B = (bjk)n x p, then the product AB is the m x p matrix C= (cik)m x p, where .

The product AB is defined only if number of columns of the matrix A is equal to the number of rows of the matrix B.

Example 1. Multiply the matrices

Example 2. Evaluate the square of the matrix

The set of all square matrices of a given order n has a multiplicative identity, that is, there is a unique n x n matrix In such that AIn = InA = A, for any n x n matrix A. We say that In is the identity matrix of order n, or simply, the identity matrix. It can be shown that each entry on the main diagonal of In is 1 and all other entries are 0:

Properties
A(BC) = (AB)C | Associative Law
A(B + C) = AB + AC, (A + B)C = AC + BC | Distributive Laws
AIn = InA = A | Multiplication Identity

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