12-3 Determinants and Inverses
Quick Review
A square matrix with 1's along its main diagonal and 0's elsewhere is the multiplicative identity matrix, I. If A and X are square matrices such that
A
X
=
I
,
eh x equals i comma then X is the multiplicative identity matrix of
A
,
A
−
1
.
eh comma , eh super negative 1 end super , .
You can use a calculator to find the inverse of a matrix. You can find the inverse of a
2
×
2
2 times 2 matrix
A
=
[
a
b
c
d
]
eh equals . matrix with 2 rows and 2 columns , row1 column 1 , eh , column 2 b , row2 column 1 , c , column 2 d , end matrix by using its determinant.
A
−
1
=
1
det
A
[
d
−
b
−
c
a
]
=
1
a
d
−
b
c
[
d
−
b
−
c
a
]
eh super negative 1 end super , equals . fraction 1 , over det eh end fraction . matrix with 2 rows and 2 columns , row1 column 1 , d , column 2 negative b , row2 column 1 , negative c , column 2 eh , end matrix . equals . fraction 1 , over eh d minus b c end fraction . matrix with 2 rows and 2 columns , row1 column 1 , d , column 2 negative b , row2 column 1 , negative c , column 2 eh , end matrix
Example
What is the determinant of
[
2
−
3
3
−
4
]
?
. matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 negative 3 , row2 column 1 , 3 , column 2 negative 4 , end matrix . question mark
det
[
2
−
3
3
−
4
]
=
(
2
)
(
−
4
)
−
(
−
3
)
(
3
)
=
−
8
−
(
−
9
)
=
1
table with 2 rows and 2 columns , row1 column 1 , . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 negative 3 , row2 column 1 , 3 , column 2 negative 4 , end matrix , column 2 equals , open 2 close . open , negative 4 , close . minus . open , negative 3 , close . open 3 close , row2 column 1 , , column 2 equals negative 8 minus . open , negative 9 , close . equals 1 , end table
Exercises
Evaluate the determinant of each matrix and find the inverse, if possible.
-
[
6
1
0
4
]
. matrix with 2 rows and 2 columns , row1 column 1 , 6 , column 2 1 , row2 column 1 , 0 , column 2 4 , end matrix
-
[
5
−
2
10
−
4
]
. matrix with 2 rows and 2 columns , row1 column 1 , 5 , column 2 negative 2 , row2 column 1 , 10 , column 2 negative 4 , end matrix
-
[
10
1
8
5
]
. matrix with 2 rows and 2 columns , row1 column 1 , 10 , column 2 1 , row2 column 1 , 8 , column 2 5 , end matrix
-
[
1
0
2
−
1
0
1
−
1
−
2
0
]
. matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 0 , column 3 2 , row2 column 1 , negative 1 , column 2 0 , column 3 1 , row3 column 1 , negative 1 , column 2 negative 2 , column 3 0 , end matrix
12-4 Inverse Matrices and Systems
Quick Review
You can use inverse matrices to solve some matrix equations and systems of equations. When equations in a system are in standard form, the product of the coefficient matrix and the variable matrix equals the constant matrix. You solve the equation by multiplying both sides of the equation by the inverse of the coefficient matrix. If that inverse does not exist, the system does not have a unique solution.
Example
What is the matrix equation that corresponds to the following system?
{
2
x
−
y
=
12
x
+
4
y
=
15
left brace . table with 2 rows and 1 column , row1 column 1 , 2 x minus y equals 12 , row2 column 1 , x plus 4 y equals 15 , end table
Identify
A
=
[
2
−
1
1
4
]
,
X
=
[
x
y
]
,
eh equals . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 negative 1 , row2 column 1 , 1 , column 2 4 , end matrix . comma . x equals , matrix with 2 rows and 1 column , row1 column 1 , x , row2 column 1 , y , end matrix . comma and
B
=
[
12
15
]
.
b equals . matrix with 2 rows and 1 column , row1 column 1 , 12 , row2 column 1 , 15 , end matrix . .
The matrix equation is
A
X
=
B
eh x equals b or
[
2
−
1
1
4
]
[
x
y
]
=
[
12
15
]
.
. matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 negative 1 , row2 column 1 , 1 , column 2 4 , end matrix , matrix with 2 rows and 1 column , row1 column 1 , x , row2 column 1 , y , end matrix . equals . matrix with 2 rows and 1 column , row1 column 1 , 12 , row2 column 1 , 15 , end matrix . .
Exercises
Use an inverse matrix to solve each equation or system.
-
[
3
5
6
2
]
X
=
[
−
2
6
4
12
]
. matrix with 2 rows and 2 columns , row1 column 1 , 3 , column 2 5 , row2 column 1 , 6 , column 2 2 , end matrix x equals . matrix with 2 rows and 2 columns , row1 column 1 , negative 2 , column 2 6 , row2 column 1 , 4 , column 2 12 , end matrix
-
{
x
−
y
=
3
2
x
−
y
=
−
1
left brace . table with 2 rows and 2 columns , row1 column 1 , x minus y , column 2 equals 3 , row2 column 1 , 2 x minus y , column 2 equals negative 1 , end table
-
[
4
1
2
1
]
[
x
y
]
=
[
10
6
]
. matrix with 2 rows and 2 columns , row1 column 1 , 4 , column 2 1 , row2 column 1 , 2 , column 2 1 , end matrix , matrix with 2 rows and 1 column , row1 column 1 , x , row2 column 1 , y , end matrix . equals . matrix with 2 rows and 1 column , row1 column 1 , 10 , row2 column 1 , 6 , end matrix
-
[
−
6
0
7
1
]
X
=
[
−
12
−
6
17
9
]
. matrix with 2 rows and 2 columns , row1 column 1 , negative 6 , column 2 0 , row2 column 1 , 7 , column 2 1 , end matrix x equals . matrix with 2 rows and 2 columns , row1 column 1 , negative 12 , column 2 negative 6 , row2 column 1 , 17 , column 2 9 , end matrix
-
{
x
+
2
y
=
15
2
x
+
4
y
=
30
left brace . table with 2 rows and 2 columns , row1 column 1 , x plus 2 y , column 2 equals 15 , row2 column 1 , 2 x plus 4 y , column 2 equals 30 , end table
-
{
a
+
2
b
+
c
=
14
b
=
c
+
1
a
=
−
3
c
+
6
left brace . table with 3 rows and 2 columns , row1 column 1 , eh plus 2 b plus c , column 2 equals 14 , row2 column 1 , b , column 2 equals c plus 1 , row3 column 1 , eh , column 2 equals negative 3 c plus 6 , end table
