Matrix Multiplication

• Matrix multiplication is noncommutative-that is, multiplying a matrix A by a matrix B is not the same as multiplying a matrix B by a matrix A.

• Not all matrices can be multiplied: we must consider the dimensions of each matrix. if the number of columns of the first matrix is the same as the number of rows of ths second, then the matrices may be multiplied together: i.e. two matrices (m x n) and (n x p) maybe be multiplied.

Consider the matrices N, P and Q

     (a b)

N= (c d)     

      (e f)

P= (g h)

      (w x)

Q= (y z)

Matrices N and P are both (2 x 2) matrices, and so can be multiplied together. NP=Q

(a b) (e f)

(c d) (g h)

To perform this operation, we need to consider each value in the matrix Q. The value for w is obtained by considering the elements in the first row of matrix N, and the first column of matrix P. This is because w is in the first column and first row of matrix Q

w=a*e + b*g

As you can see, we multiply the first element in the first row by the first element in the first column, and add it to the product of the second element in the first row and the second element in the first column.

Because x is in the first row and the second column of Q, we take the first row of matrix N and the second column of matrix P, to get

x=a* f + b*h

Going down a row to y, we need the second row and first column:

y=c*e+d*g

And for z:

z=c*f + d*h

So matrix Q is:

      (ae+bg af+bh)

Q = (ce+dg cf+dh)

This seems somewhat complex, but when dealing with numerical values many of the calculations can be done as you work through the multiplication. If you leave it all to the end, then trying to type everything into your calculator is probably more likely to lead you to mistakes.

• Take the corresponding row from the first matrix, and the column from the second

• Multiply the corresponding elements in a row to the elements in the column, and add this to the product of the other elements (i.e. first in row, first in column summed to second in row, second in column)

• An element in the product matrix in row a column b will be taken from the elements in row a of the first matrix and column b of the second

      (3 2)

A= (1 4)

      (3 1)

B= (5 2)

If C=AB, find the value of C

      (3 2) (3 1)     (j k)

C= (1 4) (5 2) = (l m)

j=3*3 + 2*5 = 9 + 10 = 19

k=3*1 + 2*2 = 3 + 4 = 7

l=1*3 + 4*5 = 3 + 20 = 23

m=1*1 + 4*2 = 1 + 8 = 9

So matrix C:

     (19 7)

C= (23 9)