Properties of Matrix Multiplication

When two matricesandare multiplied to produce a third matrixthe entry in the ith row and jth column, labelledcan be considered as a dot product.

If the matrixis considered to be made up of row vectorsand the matrixis considered to be made up of column vectorsthen the elementinis the dot product ofwith

This view is helpful in understanding the following property of matrix multiplication.

Example

Then

The proof can be written in terms of the dot product.

If the ith row ofisand the jth column ofisthen the element in the ith row and jh column ofisWhen the transpose ofis taken, this will be the element in the jth row and ith column.

When the transpose ofis taken, the ith row will become the ith column, and when the transpose ofis taken, the jth column will become the jth row. The element in the jth row and ith column ofwill be the dot product ofwithas before hence

Another important properties of matrix multiplication concerns inverses:

The proof of this is quite easy.

An inverse ofissinceand

Also the inverse is unique since ifis any other inverse thenand

For the matrices A and B above