[Solved] 18.06 Exam

This question is about an m by n matrix A for which

1 0

^Ax has no solutions and has exactly one solution.

• Give all possible information about m and n and the rank r of A.
• Find all solutions to Ax = 0 and explain your
• 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):

• Write the inverse matrix A¹ in terms of the Es. Then compute A¹ .
• What is the original matrix A ?
• 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

• For each c find a basis for the column space of A.
• For each c find a basis for the nullspace of A.

1

• 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 ?

• 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.

• 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 ?