Ch 2-3 Determinants & Multiplicative Inverses of Matrices (Part 2)
Obj: To evaluate determinants, find inverses of matrices, and to solve systems of
equations by using inverses of matrices
Recall: Identity Matrix for Multiplication
- Square matrix
- Upper left to lower right diagonal all 1’s
- Rest of elements  0’s
Multiplicative Inverse – ANY number times its multiplicative inverse equals the
“Multiplicative Inverse”
3
Ex 1.
4
∙
=1
This implies that matrices can ALSO have inverses
Inverse of 2nd Order Matrix
𝑎 𝑏
𝑎
If 𝐴 = [
]
𝑎𝑛𝑑
|
𝑐 𝑑
𝑐
𝐴−1 =
1
|
𝑎
𝑐
𝑑
∙
[
𝑏
| −𝑐
𝑑
𝑏
| ≠ 0, 𝑡ℎ𝑒𝑛
𝑑
−𝑏
]
𝑎
OR 𝐴−1 =
Ex 2. Find the multiplicative inverse of
𝐴=[
3
4
−1
]
2
1
𝐷𝑒𝑡. 𝐴
∙[
𝑑
−𝑐
−𝑏
]
𝑎
Check…..Does 𝐴 ∙ 𝐴−1 = 𝐼 ?
1
[
1
3 −1
] ∙ [ 52
4 2
−
5
Recall above…
1) [
−1 4
]
3 2
2) [
−1 0
]
8 2
10
3]
=
10
𝐴−1 =
1
𝐷𝑒𝑡. 𝐴
∙[
𝑑
−𝑐
−𝑏
]
𝑎
Using Matrices to Solve Systems – STEPS
-1 Write system as a matrix equation
-2 Find the Inverse of the Coefficient Matrix
-3 Multiply BOTH sides by the Inverse
Given
5𝑥 + 4𝑦 = −3
3𝑥 − 5𝑦 = −24
Step 1 Matrix Equation
Try these on your own:
𝑥
−3
5 4
[
] ∙ [𝑦 ] = [
]
−24
3 −5
Step 2 Find the Inverse (𝐼) of Coefficient Matrix
Step 3 Multiply Both Sides by Inverse
1 −5 −4 5 4
1 −5 −4
𝑥
−3
− [
]∙[
] ∙ [𝑦 ] = − [
]∙[
]
−24
3 −5
37 −3 5
37 −3 5
𝐼
Coeff.
Matrix
𝐼
𝑥
[𝑦 ] =
(-3,3)
Try solving these systems using Matrix Equations…
𝑥
4 8
7
[
] ∙ [𝑦 ] = [ ]
0
2 −3
* #11 p.75
3 1
( , )
4 2
5𝑥 + 𝑦 = 1
9𝑥 + 3𝑦 = 1
1
5 1 𝑥
[
] ∙ [𝑦 ] = [ ]
1
9 3
HW 2-3 Part 2…p. 75/ #9 – 11, 19 – 25 ODDS, 26, 35
1 2
( , )
2 3
Warm Up
Find the determinant of each:
|
2 10
|
−3 7
1 2 −3
|0 −3 1 |
5 −1 4