Math 237-101
Summer 2013
Quiz 7, June 18
Carter
Name
Express the matrix
"
M=
4 7
7 9
#
as a linear combination of the matrices A, B, and C where
"
A=
#
1 1
1 1
"
,
B=
1 2
3 4
#
"
,
and C =
Solution. This is tantamount to solving the system of
below:





1 1 1
x
 1 2 1 


 



· y =
 1 3 4 

z
1 4 5
1 1
4 5
#
.
equations (written in matrix form)
4
7
7
9


1
0
0
0


.

We row reduce the augmented matrix





1
1
1
1
1
2
3
4
1
1
4
5
R2 − R1 7→ R20
R3 − R1 7→ R30
4
0
7 
 R4 − R1 7→ R4
−→

7 
9

R1 − R2 7→ R10
R3 − 2R2 7→ R30
R4 − 2R2 7→ R40
−→





1
0
0
0
0
1
0
0




1
1
2
3
1 1
0 3
3 −3
4 −4
1
0
3
4
4
3
3
5










Thus z = −1, y = 3, and consequently x = 2. We have
"
M=
4 7
7 9
#
"
=2
1 1
1 1
#
"
+3
1 2
3 4
#
"
−
1 1
4 5
#
= 2A + 3B + (−1)C.