1. What are all the possible ranks for a 3 x 3 matrix? Give an example of each. 2. Repeat the question above but use only examples with no zero entries, or else (for a given rank) explain why it's not possible to do so. 3. (a) Give a 3 x 4 augmented matrix that represents an underdetermined system. (b) Give a 3 x 4 augmented matrix that represents a uniquely determined system. (c) Give a 3 x 4 augmented matrix that represents an inconsistent system. 4. If there is a matrix A and two vectors 6, e such that the system A = 6 has a different number of solutions than the system Ar = c, what can you conclude about A? (Put another way, what does it say about a coefficient matrix when the right-hand side vector determines how many solutions a linear system has?) ] 6. Is it possible for a noninvertible 2 x 2 matrix to have exactly one zero entry? Explain. 7. (a) Do all diagonal matrices have inverses? If not, which do? (b) Explain in a sentence how to write the inverse of a diagonal matrix of any size. 5. *For which values of k is the matrix -2 k 3 k +1 invertible? [k -2 3 k +1 ? Explain. 8. (a) Give a matrix which is its own inverse. (b) *Assuming you "picked the low-hanging fruit" in (a), give another matrix of the same size which is also its own inverse. P.S. Googling is cheating here, but for more info seek idempotent matrices. 9. *Give a matrix whose square is a zero matrix. Such matrices are called nilpotent. 10. Explain why one can tell at a glance whether or not a 2 x 2 matrix is invertible, but not necessarily a 3 x 3. (And what distinguishes those 3 x 3 matrices where you can tell at a glance, from those where you can't?) 11. (a) Give a 5x5 singular matrix with no zero entries. (b) Give a 5x5 matrix whose determinant is 4. 12. Does there exist a matrix A such that det(A) = -9? If so, give an example. If not, why not? 13. If the system Ax = b has a unique solution, what can you say about the dimensions of the mx n matrix A? (Hint: think about what rank A needs to be.) 14. If the m x n matrix A is wider than it is tall, m n, what can you say about the RREF of A? (again due to the rank)