1. Determine the value of c (if any) so that the system of equations with the following augmented matrix

has the requested number of solutions. Make your rationale clear!

(a) No solution.

(b) In?nitely many solutions.

(c) Exactly one solution.

(d) Exactly two solutions.

2. Find the inverse of the following matrix by using Gauss-Jordan elimination applied to (A|I). Please show all the details.

It's a good idea to validate your answer by selecting sample values of parameters a; b; c and checking (using a calculator) that your calculations give a correct answer.

3. Using the result of problem 2, ?nd the inverse of the matrix.

4. Give an example of matrices A and B such that AB exists but BAdoesn't.

5. Give an example of a matrix A such tah A^{2} = 0. (A^{2} means AA).

6. Show that if A is an n n invertible matrix, there could be no nonzero n n matrix B such that AB = 0.

HINT: Sometimes, when you need to show that something could not happen, a good way to argue is to suppose that it could and then arrive to a logical contradiction.

7. Find elementary matrices E_{1} and E_{2}, such that U = E2E1A is upper triangular. Factor A into A = LU by ?nding inverses of E_{1} and E_{2}.

8. Consider A, L, and U from problem7, and let Solve the system

Ax = b

by solving the equation Ly = b ?rst (using substitution). Once you ?nd y, solve the equation Ux = y by back substitution.

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