Differential Equations Workbook For Dummies®

11.2. An Exercise in Muddying Your Mind: Multiplying Matrices

To effectively employ matrices when solving systems of equations, you need to have some matrix-multiplication skills in your arsenal. Multiplying matrices is a little more involved than simply adding them. Why? Because AB is defined when the number of columns in A is the same as the number of rows in B. That is, if A is an l × m (that's row × column notation, so A has l rows and m columns) matrix and B is an m × n matrix, then the product AB exists — and the product is an l × n matrix.

NOTE

If AB = C, then the (i, j) (that's row, column) element of C is found by multiplying each element of the ith row of A by the matching element in the jth column of B and then adding the resulting products. Following is the standard visual presentation of multiplying matrices:

Here's a little tidbit that may come in handy for you: AB BA.

As you multiply matrices, you may occasionally encounter something called the identity matrix. It's labeled I and holds 1s along its upper-left to lower-right diagonal; the other numbers in it are all 0s. Check out this 2×2 identity matrix to see what I mean:

A 3×3 identity matrix looks like this: ...