Description
1 (24 pts.) This question is about an m by n matrix A for which
1 0
⎡ ⎤ ⎡ ⎤
Ax = ⎢
⎣1
⎥
⎦
has no solutions and Ax = ⎢
⎣1
⎥
⎦
has exactly one solution.
1 0
(a) Give all possible information about m and n and the rank r of A.
(b) Find all solutions to Ax = 0 and explain your answer.
(c) Write down an example of a matrix A that fits the description in
part (a).
2
2 (24 pts.) The 3 by 3 matrix A reduces to the identity matrix I by the following three
row operations (in order):
E21 : Subtract 4 (row 1) from row 2.
E31 : Subtract 3 (row 1) from row 3.
E23 : Subtract row 3 from row 2.
(a) Write the inverse matrix A−1 in terms of the E’s. Then compute A−1.
(b) What is the original matrix A ?
(c) What is the lower triangular factor L in A = LU ?
4
3 (28 pts.) This 3 by 4 matrix depends on c:
⎡ ⎤
1 1 2 4
A = ⎢
⎣3 c 2 8
⎥
⎦
0 0 2 2
(a) For each c find a basis for the column space of A.
(b) For each c find a basis for the nullspace of A. ⎡ ⎤
1
(c) For each c find the complete solution x to Ax = ⎣
⎢ c ⎦
⎥.
0
6
4 (24 pts.) (a) If A is a 3 by 5 matrix, what information do you have about the
nullspace of A ?
(b) Suppose row operations on A lead to this matrix R = rref(A):
⎡ ⎤
1 4 0 0 0
R = ⎢
⎣0 0 0 1 0
⎥
⎦
0 0 0 0 1
Write all known information about the columns of A.
(c) In the vector space M of all 3 by 3 matrices (you could call this a
matrix space), what subspace S is spanned by all possible row reduced
echelon forms R ?
8
