**Instructions**. Begin by printing your name in legible block characters
at the top of your workbook. Each problem is worth 25 points. At each stage in your
work, please be sure to identify the problem (or part of problem) you’re
currently working on. Before turning your workbook in, be sure to record there,
in your own words, your adherence to the Wesleyan Honor Code.

1. For each of the following matrices, determine

(i) whether it is already in fully reduced row echelon (frre) form;

(ii) (if it is not already in frre form) what its frre form is;

(iii) what its rank is;

(iv) whether it has a determinant, and, if so, what that is; and

(v) whether it must be invertible, and, if so, what its inverse is:

**
0 1 0 1 0 0 5 0 0
1 3
***A* = 0 0 1 , *B* = 0 0 1 3 , *C* = 0 0 , *D* = .
2 7
1 0 0 0 1 0 4 0 0

2. For which values, if any, of the real number *a* will the following
system of three equations (in the unknowns *x* , *y* , and *z* ) have

(i) no solutions?

(ii) exactly one solution?

(iii) more solutions than just one, but *not* infinitely many?

(iv) infinitely many solutions?
*x* + 2*y* – 3*z* = 4 ,

3*x* – *y* + 5*z* = 2 ,

4*x* + *y* + (*a*^{2} – 14)*z* = *a* + 2 .

3. Can the 2 × 2 identity matrix **I**_{2×2} be expressed as the difference
**I**_{2×2} = *AB* – *BA*

between the two possible products *AB* and *BA* of two suitable 2 × 2 matrices
*A* and *B* ?

If so, give an explicit example; if not, give a coherent explanation of why not.

[Suggestion: Can the traces tr(**I**_{2×2}) and tr(*AB* – *BA*) be of any use here?]

4. Given an *n* × *n* matrix *A* , with known determinant det(*A*) ,
calculate the determinant of the matrix product *A*•adj(*A*)
(of *A* with the adjoint matrix adj(*A*)).

5 (for extra credit, only if you have the time). Extend your Problem 3 findings
to matrices of more general size(s) than just 2 × 2 .

Posted 4 Mar 2006