This question is about an m by n matrix A for which
⎡1 ⎤ ⎡0 ⎤
Ax has no solutions and has exactly one solution.
(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
(24 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
(28 This 3 by 4 matrix depends on c:
⎡1 1 2 4 ⎤
A
(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 .
6
(24 (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 ⎤
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
MIT OpenCourseWare http://ocw.mit.edu
18.06SC Linear Algebra
Fall 2011
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