Some Additions to Review Set 2
1. What are the two ways we defined matrix multiplication, AB? (Hint: One way was
the rwo-column rule).
2. Let A, B be given below. Form the matrix product AB, if defined:
−1 1
2 0 −1
A=
B= 2 1
1 1
0
1 2
3. Given the matrix A, B below:
1 2 3
A = −1 1 0
3 2 0
1 −1 2
1 1
B= 2
1
0 1
(a) Compute only the (2, 3) entry of AB:
(b) Compute only the (3, 2) entry of AB T :
(c) Compute B − 3I3 :
(d) Compute C23 for matrix A (that’s the (2,3) cofactor).
4. If A is the 2 × 3 matrix below, find a matrix C so that AC = I, but note that C is not
the inverse of A. To simplify your computations, I’ve given you one form for C that
you might use.
c11 c12
−1
2 −1
A=
C = c21 c22
6 −9
3
0
0
5. Suppose A is n×n with the property that Ax = ~0 has only the trivial solution. Without
using the invertible matrix theorem, explain directly why the equation Ax = b must
have a solution for every b.
6. Explain why the columns of A2 span IRn whenever the columns of A are linearly
independent. (Hint: You might think about whether or not A2 must be invertible).
7. Suppose subspace H is the span of the two vectors below in set B:
2
1
0 , 3 = {v1 , v2 }
B=
0
0
(a) Does B span IR3 ? Why or why not?
(b) Find [v1 ]B
(c) If c = (3, 3, 0), find [c]B
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