Basics of Matrix Arithmetic

You need to be able to do 3 things which are illustrated below:

1) row times column = number (same as dot product)

1xn * nx1 = 1x1 or a single number

2) matrix times column = repeated applications of row times column = column

nxm * mx1 = nx1

3) matrix times matrix = repeated apps of matrix times column = matrix.

nxm * mxr = nxr

Example:

Comments:

If you don’t have the same number of entries in the columns on the left and the rows on the right then some numbers have nothing to match up to. The matrix product is simply not defined and the matrices are incompatible.

Important Example:

what does this show?? That with matrix multiplication, order matters! In general, it is not true or cannot be assumed that AB = BA. More formally, matrix multiplication is not commutative.