3.8: Use Inverse Matrices to Solve Linear Systems Objectives Assignment:

3.8: Use Inverse Matrices to Solve Linear
Systems
1.
2.
3.
Objectives
To find the inverse
of a square matrix
To solve a matrix
equation using
inverses
To solve a linear
system using
inverse matrices
Assignment:
• P. 214-217: 2, 3-42 M3,
43, 47, 51-53
• Challenge Problems
• Cryptography
Warm-Up
Imagine you lived in a world that never defined
subtraction or division. What would you do to
get the same results?
Objective 1
You will be able to find the
inverse of a square matrix
Identity Matrix
The 𝑛 × 𝑛 identity matrix has 1s along the
main diagonal and zeros everywhere else.
Identity Matrix
Recall that multiplying a number by 1 gives
you back the same number. This is the
Identity Property of Multiplication.
Similarly, multiplying a matrix by the identity
matrix, 𝐼, will return the original matrix.
Identity Property of Matrix
Multiplication
𝐴𝐼 = 𝐼𝐴 = 𝐴
Exercise 1
−1 2
−3 −4
Find 𝐴𝐵: 𝐴 =
and 𝐵 = 1 − 3
−1 −2
2
2
Inverse Matrix
The inverse of a square matrix 𝐴 is
another square matrix 𝐵 such that
𝐴𝐵 = 𝐼 and 𝐵𝐴 = 𝐼.
The inverse of
matrix 𝐴 is
denoted as 𝐴−1
The inverse is only defined
for a square matrix with a
𝑛𝑜𝑛𝑧𝑒𝑟𝑜 determinant
Exercise 2
Find 𝐴 . What is the relationship between
𝐴, 𝐵, and 𝐴 ?
−3
𝐴=
−1
−1 2
−4
and 𝐵 = 1 − 3
−2
2
2
Inverse Matrix (2x2)
Exercise 3
Find
𝐴−1 :
2
𝐴=
1
4
3
Exercise 4
Find
𝐴−1 :
6
𝐴=
2
1
4
Matrix Equations
In general a matrix equation can be written
as 𝐴𝑋 = 𝐵, where 𝐴, 𝐵, and 𝑋 are
matrices.
𝐴𝑋 = 𝐵
To solve this equation for 𝑋, you would
ordinarily divide by 𝐴. However, there is
no matrix division. Instead you solve for 𝑋
by multiplying both sides of the equation by
the inverse of 𝐴.
Matrix Equations
In general a matrix equation can be written
as 𝐴𝑋 = 𝐵, where 𝐴, 𝐵, and 𝑋 are
matrices.
𝐴𝑋 = 𝐵
How to
−1
−1
𝐴 𝐴𝑋 = 𝐴 𝐵
solve a
matrix
𝐼𝑋 = 𝐴−1 𝐵
equation
−1
𝑋=𝐴 𝐵
Since matrix multiplication is not commutative, you must
remember to always multiply on the same side.
Exercise 5
Solve the matrix equation for 𝑋.
3 −2
−2
𝑋=
−7 5
3
4
−1
Exercise 6
Solve the matrix equation for 𝑋.
−4 1
8 9
𝑋=
0 6
24 6
You will be able to
solve a linear system
using inverse
matrices
Objective 3
Exercise 7
Use an inverse matrix to solve the linear
system.
4𝑥 + 𝑦 = 10
3𝑥 + 5𝑦 = −1
Solving Systems using Inverses
Write the
system as a
Step 1
matrix
equation
𝐴𝑋 = 𝐵
Find the
inverse
Step 2
of
matrix 𝐴
Multiply both
left sides of
Step 3
𝐴𝑋 = 𝐵 by
𝐴−1
𝐴 is the coefficient matrix
The solution is 𝑋 = 𝐴−1 𝐵
𝐵 is the constant matrix
Protip: Multiplying by Inverse
When multiplying by an inverse matrix to solve a
matrix equation, first multiply by the “subinverse” and then divide each element by the
determinant.
4 1 𝑥
10
=
3 5 𝑦
−1
1 divided by the
determinant
𝑥
4
=
𝑦
3
1
𝑥
𝑦 = 17
1
𝑥
𝑦 = 17
1
5
−1
10
−1
5 −3 10
−1 4 −1
3
51
=
−2
−34
“Subinverse”
Exercise 8
Solve each system using inverse matrices.
1. 3𝑥 − 𝑦 = −5
−4𝑥 + 2𝑦 = 8
2.
2𝑥 − 𝑦 = −6
6𝑥 − 3𝑦 = −18
Exercise 8
Solve each system using inverse matrices.
1. 3𝑥 − 𝑦 = −5
−4𝑥 + 2𝑦 = 8
Exercise 8
Solve each system using inverse matrices.
2.
2𝑥 − 𝑦 = −6
6𝑥 − 3𝑦 = −18
You will be able to
solve a linear system
using inverse
matrices
Objective 3
Inverse Matrix (3x3)
𝑎
The inverse of a matrix 𝐴 = 𝑑
𝑔
A1
 e

 h
1  d

 g
A 
 d
 g

f
i
f
i
e
h
b c
h i
a c
g i
a b

g h

𝑏
𝑒
ℎ
b
e
a

d
a
d
𝑐
𝑓 is
𝑖
c
f
c
f
b
e









Inverse Matrix (3x3)
𝑎
The inverse of a matrix 𝐴 = 𝑑
𝑔
just use a calculator!
𝑏
𝑒
ℎ
𝑐
𝑓 is
𝑖
Exercise 9
Find 𝐴−1 .
1 −1 0
𝐴 = 1 0 −1
6 −2 −3
Exercise 10
Find 𝐴−1 .
2 1 −2
𝐴= 5 3 0
4 3 8
Exercise 11
Solve the system using inverse matrices.
2𝑥 + 3𝑦 + 𝑧 = −1
3𝑥 + 3𝑦 + 𝑧 = 1
2𝑥 + 4𝑦 + 𝑧 = −2
Exercise 12
Solve the system using inverse matrices.
𝑥 + 3𝑦 − 5𝑧 = −3
𝑥 − 2𝑦 − 𝑧 = 12
𝑥 + 3𝑦 − 𝑧 = 18
3.8: Use Inverse Matrices to Solve Linear
Systems
1.
2.
3.
Objectives
To find the inverse
of a square matrix
To solve a matrix
equation using
inverses
To solve a linear
system using
inverse matrices
Assignment
• P. 214-217: 2, 3-42
M3, 43, 47, 51-53
• Challenge
Problems
• Cryptography
“The matrix has you now.”