Goals
We will discover what the following have in common.
Existence of solutions to A~x = ~b.
Spans of vectors
Linear independence
Inverse matrices
and more
Directions
Complete the task for each section.
The explanations requested for each task have already been
covered (previous lessons).
Comparing these explanations will provide evidence the
theorem is true.
Similar Problems
Determine which of the following augmented matrices (linear
systems) have no, one, or multiple solutions. Explain why.
 

1 0 0
2
1 0 −3 2
 0 1 0 −5   0 1
6 11 
0 0 1
0 0
0 0
6




1 0 −7 9
 0 1 −2 6 
0 0
0 3
 
 

2
1 0 0
1 0 −3 2
1 0 −7 9
 0 1 0 −5   0 1
6 11   0 1 −2 6 
3
1 1 1
1 1
3 13
1 1 −9 18
Similar Problems
For each of the following matrices determine which have solutions
to A~x = ~b. Determine which matrices have an inverse. Explain
why they do or do not have an inverse.

1
 0
0

1
 0
1
~
b 
2
 −5 
6
A
0
1
0

0
0 
1


0
1
1

0
0 
1



2
 −5 
3
1
 0
0
1
 0
1
~
b 
2
 11 
0
A
0
1
0

−3
6 
0


0
1
1

−3
6 
3



2
 11 
13
1
 0
0
1
 0
1
A
0
1
0

−7
−2 
0
0
1
1

−7
−2 
−9
~
 b 
9
 6 
3



9
6 
18
Similar Problems
For each of the following matrices determine whether (0, 0, 0) is
the only solution to A~x = ~0. Explain why or why not.

 
 

1 0 0
1 0 −3
1 0 −7
 0 1 0   0 1
6   0 1 −2 
0 0 1
0 0
0
0 0
0

 
 

1 0 0
1 0 −3
1 0 −7
 0 1 0   0 1
6   0 1 −2 
1 1 1
1 1
3
1 1 −9
Similar Problems
For each of the following matrices determine whether A~x = ~b has
at least one solution for all ~b. Explain why or why not.

 
 

1 0 0
1 0 −3
1 0 −7
 0 1 0   0 1
6   0 1 −2 
0 0 1
0 0
0
0 0
0

 
 

1 0 0
1 0 −3
1 0 −7
 0 1 0   0 1
6   0 1 −2 
1 1 1
1 1
3
1 1 −9
Similar Problems
For each of the following matrices determine whether the columns
span R3 (the set of all three entry vectors). Explain why or why
not.

 
 

1 0 0
1 0 −3
1 0 −7
 0 1 0   0 1
6   0 1 −2 
0 0 1
0 0
0
0 0
0

 
 

1 0 0
1 0 −3
1 0 −7
 0 1 0   0 1
6   0 1 −2 
1 1 1
1 1
3
1 1 −9
Similar Problems
For each of the following matrices determine whether the columns
are linearly independent. Explain why or why not.

 
 

1 0 0
1 0 −3
1 0 −7
 0 1 0   0 1
6   0 1 −2 
0 0 1
0 0
0
0 0
0

 
 

1 0 0
1 0 −3
1 0 −7
 0 1 0   0 1
6   0 1 −2 
1 1 1
1 1
3
1 1 −9
Similar Problems
If a matrix A has an inverse, will it be the identity matrix after row
reduction?