Math 22A
UC Davis, Winter 2011
Prof. Dan Romik
Midterm Exam 2 Solutions
1.
For each of the following matrices, determine if it is invertible, and if it is, ﬁnd
its inverse matrix.
(a)
±
1 2
3 4
²
(c)
1 1
1
0 2

2
1 3

1
(b)
1 0
2
0 4

2
0 0
0
(d)
1 1 1
0 1 2
0 1 3
Solution.
(a) Recall that the inverse of a 2
×
2 matrix
±
a b
c d
²
exists if
its determinant
ad

bc
6
= 0, and in that case the inverse matrix is equal to
1
ad

bc
±
d

b

c
a
²
. In this case
ad

bc
= 4

6 =

2 so the inverse matrix
exists and is
±

2
1
3
2

1
2
²
.
(b) This matrix has a row of zeroes so its determinant is 0 and therefore it is
not invertible.
(c) The determinant of this matrix is also 0 so it is not invertible.
(d) By applying elementary row operations to bring the matrix to reduced row
echelon form and performing the same operations in parallel on the identity
matrix, we get
1 1 1
1 0 0
0 1 2
0 1 0
0 1 3
0 0 1
∼
1 0

1
1

1 0
0 1
2
0
1
0
0 0
1
0

1 1
∼
1 0 0
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 Winter '08
 chuchel
 Linear Algebra, Algebra, Matrices

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