MATRIX
A1 =
A2 =
NULL SPACE
4 −1
−8 2
1 5
2 2


1 0


A3 =  −1 1 
1 0
A4 =
A5 =
4 3 1
0 1 1





A9 = 





A10 = 
0
0




α



1 2 −1 3
2 4 −2 6
3 6 −3 9
1 2 −1 3










α








α



−2
1
0
0

2
1
0
0






+β







+β




1


α  −1 
−1
0


 0 
0




1 −2 0 −1
−2 4
1
2
1 −2 −1 −1
1 −2 3 −1
2
3
1
0




α 2 + β 1 
−2
1
1
0




α 0 +β 1 
1
−2


R2
−2


α  −1 
1


1
0




α  −1  + β  1 
1
0


−1 2 1


A8 =  1 5 4 
3 1 −2



−1 2
1


A7 =  1 −2 −1 
1 −2 −1
R2
1
−2




α 0 +β 1 
1
0

1
−2
0
0
1


α  −2 
2
1 0 2


A6 =  2 1 5 
−2 1 −3

α

2 4 −2
3 6 −3
1
4
α
RANGE
1
0
1
0

R3
1
0
0
1







α





+γ




−3
0
0
1
1
−2
1
1






+β








α



1
2
3
1





0
1
−1
3





True–False, circle correct answer, or give a brief answer.
T–F
(1) There exists a solution to A1 x =
3
2
.
T–F
.
(2) There is a unique solution to the system of linear equations
T–F
(3) The general solution to A2 x =
T–F
T–F
6
4
is α
1
1
.
x + 5y = 0
2x + 2y = 0



x + 2z
=0
(4) There is a unique solution to the system of linear equations
2x + y + 5z = 0 .


−2x + y − 3z = 0


1


(5) There is a unique solution to A8 x =  2  .
3
(6) The range of A7 is a point, line, or plane.
(7) The null space of A5 is a point, line, or plane.
T–F
T–F
T–F
T–F
(8) The equation A3 x = b has a solution
for
every b ∈ R3 .


1


(9) There exists a solution to A3 x =  0 .
1


1


(10) There exists a unique solution to A3 x =  0  .
1
(11) There exists a unique solution to A3 x = b for every b ∈ R3 .
T–F
(12) There exists a solution to A5 x =
4
6
.
4
6
T–F
(13) There exists a unique solution to A5 x =
T–F
(14) The equation A4 x = b has a solution for every b ∈ R2 .
T–F
(15) The equation A4 x = b has a unique solution for every b ∈ R2 .
T–F
(16) The general solution to A4 x =
8
2

.



1
1



is 
1
+
α
−2



.
1
2
(17) Give two independent (i.e. non-parallel) solutions to




(18) Find the general solution to A9 x = 
1
−1
0
4



 −x + 2y + z
=0
x − 2y − z = 0 .


x − 2y − z = 0


.




x+2 z
=0
2x + y + 5z = 2
(19) Find the general solution to


−2x + y − 3 z = 2
T–F
(20) Circle the matrices which have inverses: A1 , A2 , A6 , A7 , A8 , A9 , A10 .


1
 −2 


 is a point, line, plane, or a 3-space.
(21) The solution space for A9 x = 
 1 
1


2
 4 


(22) The solution space for A10 x =   is a point, line, plane, or a 3-space.
 6 
2
(23) There exists a solution to A10 x = b for every b ∈ R4 .





T–F
.
(24) The system 



of bi ’s.
x1 − 2x2 − x4
−2x1 + 4x2 + x3 + 2x4
x1 − 2x2 − x3 − x4
x1 − 2x2 + 3x3 − x4
= b1
= b2
has a solution for all choices
= b3
= b4
(25) Find a vector perpendicular to the null space of A5 .



(26) Explain why 




(27) Explain why 

2
1
1
4



 cannot be in the null space of A9 .

22
−64
42
−38



 is in the range of A9 .
