An important observation to make is that the computation of the net result of the two successive transformations involved starting with the two matrices that define the two individual transformation and ending up with a single equivalent transformation matrix. That is, we computed

This combining of two matrices into one is called ''multiplication'' of matrices. That is, the matrix that characterizes the equivalent transformation can be computed by "multiplying" the second transformation matrix by the first transformation matrix.

You can see from the example above that matrix multiplication is NOT simply multiplying the upper left entry in the first matrix by the upper left entry in the second to get the upper left entry in the resulting matrix. Instead, the rules of multiplication of two matrices in general are defined as follows:

This can be checked by computing first the transformation

The second transformation performed on these transformed coordinates then gives

Check by hand that the example matrix multiplication above satisfies the formula for matrix multiplication.

Practice
Matrix Multiplication

Then, try out matrix multiplication on a few samples provided in the applet below.

This applet allows the user to practice matrix multiplication with 2x2 matrices. Enter the entries in their corresponding locations in the resulting matrix and press "Submit". If the answer is incorrect the user will be given two additional attempts after which the correct answer will be displayed. Press "Next" to try a different example. Press "Clear" to clear all entries.