How do I multiply two matrices together?

In order to multiply two matrices together, you must first have that the number of columns in the first matrix must be equal to the the number of rows in the second matrix.

So if you wish to multiply matrix A and matrix B,

matrix A must have dimensions ( m x n ) whilst matrix B must have dimensions ​​(​n ​x q) . Once you multiply these matrices, the new matrix formed will have dimension m xq​ .

(Note that m and q can be equal)

For example, matrix A can have dimension 3x2 and matrix B can have dimension 2x4 and so the new matrix will have dimension 3x4.

To multiply matrices, you take the first row of Matrix A and multiply its elements by the elements in the first column of Matrix B. You then take the first row of Matrix A and multiply its elements by the elements in the second column of Matrix B. You then keep repeating this process till you have multiplied the elements of the first row of Matrix A to the elements of  every single column of Matrix B.

You then take the second row of Matrix A and multiply its elements by the elements of the first column of Matrix B. You then take the second row of Matrix A and multiply it by the elements of the second column of Matrix B. You keep doing this till you have multiplied the second row of Matrix A by every single column of matrix B.

You then repeat this process for the third, fourth, fifth (and so on...) row of matrix A.  The following example should make this clear.

EXAMPLE

​​Matrix A is ( A B )    and Matrix B is ( 1 2 3 4 )

                     ( C D )                                   ( 5 6 7 8) 

The dimension of A is 2x2 and the dimension of B is 2x4. Hence, the dimension of the new matrix shall be 2x4.

Using the above rules, it should be simple to follow that new matrix is:

[ (1a+5b)   (2a+6b)   (3a+7b)  (4a+8b) ​]

[ (​1c+5d)   ​(​2c+6d)​   (3c+7d)   (4c+8d) ​]

Also note that matrix A multiplied by matrix B does not yield the same result as matrix B multiplied by matrix A. The order in which you multiply matrices matters.