8.9. Inverse Matrices http://www.ck12.org
1 2 3
1 0 1
0 2 − 1
·
x
y
z
=
96
36
− 12
Solution:
A·
x
y
z
=
96
36
− 12
A−^1 ·A·
x
y
z
=A−^1 ·
96
36
− 12
x
y
z
=A−^1 ·
96
36
− 12
x
y
z
=
−^134313
(^16) − (^1613)
(^13) − (^13) − (^13)
·
96
36
− 12
x
y
z
=
−^13 · 96 +^43 · 36 +^13 ·(− 12 )
(^16) · 96 − (^16) · 36 + (^13) ·(− 12 )
(^13) · 96 − (^13) · 36 − (^13) ·(− 12 )
x
y
z
=
12
6
24
Example C
Find the inverse of the following matrix.[
1 6
4 24
]
Solution:[
1 6
4 24
∣∣
∣∣1 00 1
] →
→ − 4 I →
[1 6
0 0
∣∣
∣∣−^1 4 1^0
]
This matrix is not invertible because its rows are not linearly independent. To test to see if a square matrix is
invertible, check whether or not the determinant is zero. If the determinant is zero then the matrix is not invertible
because the rows are not linearly independent.
Concept Problem Revisited
Non-square matrices do not generally have inverses. Square matrices that have determinants equal to zero do not
have inverses.
Vocabulary
Multiplicative inversesare two numbers or matrices whose product is one or the identity matrix.
Guided Practice
- Confirm matrixAandA−^1 are inverses by computingA−^1 ·AandA·A−^1.