Some basic terms:

* I*th row and *j*th column

*m* row by *n* column matrix

When multiplying matrices multiply row by column.

Using a third variable *p* to describe the row or column in a pair of matrices to be multiplied

A = m x p

B = p x n

A(B) = m x n

So what I just said is that *m* rows by *p* columns when multiplied by a matrix which has an equal number of rows *p* as the columns of the original matrix, will give an answer of a matrix which has as many rows as matrix *A* and as many columns as matrix *B*.

So For example, we have two matrices. *A* and *B*

*A* has 3 rows and 2 columns

*B* has 2 rows and 4 columns.

We can multiply!

And the answer should have 2 rows and 4 columns. We can multiply if *p* is equal in both matrix A and B.

What happens if *p* is not equal in both matrices?

Then *AB* is undefined

Practice Problems from HotMath

**Problem 19:*** Is the matrix product of the given matrices defined? If so, find it. If not, give reasons.*

Reasoning:

Observing the matrices shows *p* to be equal, giving us an *m x n* matrix of 2×1

[2 x 3][3×1] = [2×1]

So what exactly did I do?

First step was to lay the *B* matrix on itâ€™s side to equal the number of rows in matrix *A* that is what gives me the 2 rows part. Since I have only one column to work with in *B* I only lay that column down, but if I had two columns I would have laid it down twice per row.

Then I drop the rows from matrix *A* down to match up.