Matrix Multiplication

Matrix multiplication is:

• NOT commutative -
• associative -
• distributive -

To multiply a matrix by another matrix the number of columns in the first matrix must be equal to the number of rows in the second matrix. That’s because, when we multiply one matrix by another, we multiply the rows in the first matrix by the columns of the second matrix.

The easy way to tell whether or not you can multiply matrices is to line up their dimensions.

If the middle numbers match like they do here (they’re both ), then you can multiply the matrices to get a valid result.

To get the dimensions of the product, take the rows of the first matrix and the columns of the second matrix:

Dot product §

For instance, we said before that when we take the product for

the first entry we’ll need to find is the dot product of the first row in and the first column in . The first row in is the ordered pair , and the first column in is the ordered pair .

To take the dot product of these ordered pairs, we take the product of the first values, and then add that result to the product of the second values. In other words, the dot product of and is

The product of two matrices looks like this:

Therefore, to find the product of matrices and , we get

Zero matrix §

Multiplying a matrix by the Zero matrix, results in a zero matrix, but might have different dimensions.