**Be sure to study**
Solutions to Problem Sets 1-3 that are up on ASULearn, along with the practice
problem solutions,
and make sure that you understand the related
material (statements of theorems, examples of false statements, ...)
very well.

--The number of solutions of a system of

Specifically, what were the examples we used [with "dots" in them to account for the number of rows being unknown

In addition, how did the heart of the proof work? Ie how did we show that if we started with 2 solutions, then really we had infinitely many solutions that we generated?

For example, you might be given any of the true/false statements, such as:

1) If C is not invertible and AC=BC then A=B sometimes but not always. (True)

2) If C is invertible and AC=BC then A=B sometimes but not always. (False - it is always true since you can multiply by the inverse of C on the right.)

3) As long as the matrix mult is defined, then (A-B)(A+B) always equals A^2 - B^2. (False because A^2-BA+AB-B^2 is sometimes but not always the same as A^2-B^2, even for invertible matrices, by examples)

4) Matrix multiplication is always commutative (ie AB=BA) for square matrices. (False because it is sometimes but not always true by examples).

5) For any square matrices A and B of the same size, if AB = 0 then A=0 or B=0. (False there are lots of counterexamples).

6) If A is an invertible

True/False statements and counterexamples has been a recurring theme in all of the chapters, so you can expect problems like we have seen before in practice problems and problem sets. Be sure that you know how to find counterexamples, or write these down on your cheat sheet.

You should also know some of the basic visualizations of rows of augmented matrices from Chapter 1 - that a linear system in two variables with one free variable [x+y=1] is a line, and that a linear system in three variables [x+y+z=1] with two free variables is a plane, as well as some of the visualizations of intersections of these to form solutions, like on ASULearn. Recall that once we solve an augmented matrix by reducing, the solutions tells us the geometry of the intersections of the rows: one free t variable means the rows intersected in a line, two free variables, like s and t, means the rows intersected in a plane, and no free variables arises from the rows intersecting in 1 solution, or from them being parallel, with no solutions.

Be careful to follow all directions and to read each problem very carefully to make sure you answer correctly. You do NOT need to write in complete sentences like you do on the problem sets. Show work and briefly explain where indicated. Even if it is not indicated, you may wish to do so because partial credit can be obtained for an incorrect answer as long as there is some evidence of understanding. You may use your calculator or Maple unless the problem says "by hand". When you are finished, turn in your cheat sheet with your test. You will not turn in any Maple work.

**Some Maple Commands**
Here are some Maple commands you should be pretty familiar with by now
for this test - you may wish to put these on your cheat sheet.
**
> with(LinearAlgebra): with(plots):
> A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);
> ReducedRowEchelonForm(A);
> GaussianElimination(A); ** (only for matrices with

> B:=MatrixInverse(A);

> A.B;

> A+B;

> B-A;

> 3*A;

> A^3;

> evalf(A^100);

> Determinant(A);

> implicitplot({2*x+4*y-2,5*x-3*y-1}, x=-1..1, y=-1..1);

> implicitplot3d({x+2*y+3*z-3,2*x-y-4*z-1,x+y+z-2},x=-4..4,y=-4..4,z=-4..4);