Math 54. Solutions to Sample First Midterm
1.
(10 points) Find the inverse of the matrix
A
=
1
2
3
2
6
7
1
1
2
, if it exists.
Use the
algorithm introduced in Chapter 2.
The algorithm uses row reduction of the matrix [
A
I
] :
1
2
3
1
0
0
2
6
7
0
1
0
1
1
2
0
0
1
∼
1
2
3
1
0
0
0
2
1

2
1
0
0

1

1

1
0
1
∼
1
2
3
1
0
0
0

1

1

1
0
1
0
2
1

2
1
0
∼
1
2
3
1
0
0
0

1

1

1
0
1
0
0

1

4
1
2
∼
1
2
0

11
3
6
0

1
0
3

1

1
0
0

1

4
1
2
∼
1
0
0

5
1
4
0

1
0
3

1

1
0
0

1

4
1
2
∼
1
0
0

5
1
4
0
1
0

3
1
1
0
0
1
4

1

2
Therefore the inverse is

5
1
4

3
1
1
4

1

2
.
2.
(10 points) A matrix
A
and an echelon form of
A
are given here:
A
=
1
2

1
1

1

2

4
3

3
0
1
2

3
3
3
1
2

2
2
1
∼
1
2

1
1

1
0
0
1

1

2
0
0
0
0
0
0
0
0
0
0
.
(a).
Write the solution set of the homogeneous system
A~x
=
~
0 in parametric
vector form (i.e., as a linear combination of fixed vectors, in which the weights are
allowed to take on arbitrary values).
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 Spring '08
 Chorin
 Math, Linear Algebra, −1, Col A