Pick a diagonal matrix and give it the name A. (Use the method described in Chapter 2, Generating examples using Matlab functions, for creating diagonal matrices.)
The Matlab function "diag" makes it easiest to create square diagonal matrices, so you may as well begin with a square A, say one that is 3x3. Moreover, pick an A in which the diagonal entries are all nonzero. Now create a matrix X with 3 rows (start with an X with one column) and enter the Matlab command Y = A*X. The problem of division on the left by A is simply the problem of recovering X given A and Y. You should try to forget what your X was and reconstruct it by looking at A and Y. You should see that this is very easy to do. Do several examples, always forgetting X, until you see how to find X using just A and Y. Notice that the number of columns X has is irrelevant; you just apply the method for an X with one column to each column of X separately.
Try doing the same thing for a 4x4 diagonal A (no zeros on the diagonal) and for a 2x2 diagonal matrix.
Now try a 3x3 diagonal matrix A in which one or two of the diagonal entries are zero. Note that you can no longer say with certainty what X was if you only know A and Y = A*X. If A has one diagonal zero and X is 3x2, how many entries of X can be determined if A and Y are known?
Now experiment with diagonal matrices A that are wide (more columns than rows) or high (more rows than columns). Note that these cases are quite different. Always, the problem is, given just A and Y = A*X, what can you say about X?
Once you have done enough examples that you feel you understand how to say all there is to say about X when A and Y are given (A being a diagonal matrix) reread the explanation in Section 2 (pp. 26-27). Chances are, you will have gotten a much better grasp of the solution by doing several examples using Matlab than you would have been able to get by rereading this section any number of times.
Return to Chapter 3.