Download Math 220 Written Assignment 5 Name: ID#: Circle your section: Math

Math 220 Written Assignment 5
Circle your section:
Name:
Math 220-01 (10:35-11:25)
ID#:
Math 220-02 (1:25-2:15)
I prefer that you submit your answers on a printed copy of this document, like it’s a quiz or exam. However, you
may instead rewrite the questions by hand before solving them. Staple sheets together, in order. Be neat. Always
give enough work and clear explanation so that fellow students could follow what you did (from start to finish)
just by reading your paper.
1. Suppose matrix A is m×n, matrix B is n×p, and the second column of matrix B is all zeros. What, if anything,
can you say about the second column of AB? Explain.
2. Suppose matrix A is m×n and B is n×p. Prove that if the columns of B are linearly dependent, then the
columns of AB are also linearly dependent.
Hint: Try starting with this: If the columns of B are linearly dependent, then Bx = 0 has a nontrivial solution.
3. Suppose A and B are both n×n. Prove that if A and AB are invertible, then B must be invertible.
Hint: Since you’re told A and AB are invertible, you’re allowed to play with combinations of A, A−1 , AB, and (AB)−1 .
Try to come up with such a combination, which I’ll call D, for which you can show that D−1 = B.
4. Suppose that for a particular matrix A, there is another matrix D such that AD = Im . Prove that for this
particular A, the equation Ax = b must be consistent for every b ∈ Rm .
Hint: Think about the equation ADb = b.