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):

E_{21} : |
Subtract 4(row 1) from row 2. |

E_{31} : |
Subtract 3(row 1) from row 3. |

E_{23} : |
Subtract row 3 from row 2. |

- Write the inverse matrix
*A*^{−1}in terms of the*E*’s. Then compute A^{−}.^{1} - 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*?

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