How do I know I can multiply two matrices and if so, how do I do it?

Firstly you need to check if the matrices conform. Matrices are an array of elements within a pair of brackets, there are some number of columns, usually denoted m, and some number of rows, denoted n. These two numbers, m and n, give the order of the matrix, how large it is, suppose you had a 2x3 matrix, then it has two columns and 3 rows. Matrix multiplication can only occur if the two matrices conform, that is given two matrices A and B, the operation AB (AxB) can only occur if the number of rows of B match the number of columns of A. So if A is a mx2 matrix, for AB to exist, B must be some 2xn matrix. An interesting thing to note, m and n in this example can be any number and the matrix produced by the multiplication will be mxn.

The multiplication process itself initially looks abstract and at times in unwieldy but it is relatively simple, it is also quite difficult to explain without an example. Let A and B be conforming matrices such that AB exists, to perform the operation you firstly go across the first row of A and down the first column of B, then take the sum of all corresponding pairs of elements, that is the number that is first in the row of A multiplied by the number that is first in column of B and so on for the second, third etc. In this case the number you get is the element that is in the first row and first column for AB. You then repeat this with each pair of rows in A and columns in B, so say you did this with the second row of A and the third column of B, then you get the element that is in the second row and third column of AB. Then your final result should be the matrix AB.