## Properties of Matrix Multiplication

When two matrices and are multiplied to produce a third matrix the entry in the ith row and jth column, labelled can be considered as a dot product.

If the matrix is considered to be made up of row vectors and the matrix is considered to be made up of column vectors then the element in is the dot product of with 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 of is and the jth column of is then the element in the ith row and jh column of is When the transpose of is taken, this will be the element in the jth row and ith column.

When the transpose of is taken, the ith row will become the ith column, and when the transpose of is taken, the jth column will become the jth row. The element in the jth row and ith column of will be the dot product of with as before hence Another important property of matrix multiplication concerns inverses: The proof of this is quite easy.

An inverse of is since and Also the inverse is unique since if is any other inverse then and For the matrices A and B above   