1. Consider the matrices

A = 3 4 1 B = 8 1 5 C = 3 4 1
2 -7 -1 2 -7 -1 2 -7 -1
8 1 5 3 4 1 2 -7 3

Is it possible to find an elementary matrix E such that EB = C? Justify your answer in a proof.

2. Write the matrix 3 -2 as a product of elementary matrices.

3 -1

3. Show that if A = 1 0 0 is an elementary matrix, then at least one entry in the

0 1 0 third row must be a zero. Do answer in a proof.

a b c

4. Prove that if A is an invertible matrix and B is row equivalent to A, then B is also invertible. Show answer in a proof.

5. Find the conditions that the b's must satisfy for the system to be consistent.

6x1 -4x2 = b1
6x1 -4x2 = b2

6. Consider the matrices A = 2 1 2 and x = x1

2 2 -2 x2
3 1 1 x3

a) Show that the equation Ax = x can be rewritten as (A - I)x = 0 and use this result to solve Ax = x for x.

b) Solve Ax = 4x