MATH 205A,B - LINEAR ALGEBRA
WINTER 2013
QUIZ 5
NAME: Section:(Circle one) A(1 : 10) B(2 : 40)
Show ALL your work CAREFULLY.
(a) Suppose det
a b c
d e f
g h i
= 3. Find det
a d g
2b2e2h
c f i
.
det
a d g
2b2e2h
c f i
= 2 ·det
a d g
b e h
c f i
= 2 ·det
a d g
b e h
c f i
T
= 2 ·det
a b c
d e f
g h i
= 6.
(b) Use row operations to find det Awhere
A=
1a b +c
1b a +c
1c a +b
.
Is Ainvertible?
First note that if a=bor a=cthen Awill have two identical rows and thus is row
equivalent to a matrix that has a row of zeros. In such cases, det A= 0 and Ais not
invertible. Now, assume that a6=band a6=c. Then we have
A∼
1a b +c
0b−a a −b
1c a +b
∼
1a b +c
0b−a a −b
0c−a a −c
∼
1a b +c
0 1 −1
0c−a a −c
∼
1a b +c
0 1 −1
0 1 −1
∼
1a b +c
0 1 −1
0 0 0
.
It follows that det A= 0 and hence Ais not invertible.
Date: February 13, 2013.
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