# Multiplication of Matrices

Published on: Fri Jan 23 2009

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 2x1
[2 x 3][3x1] = [2x1]
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.