Transcript 3-8 Solving Systems of Equations Using Inverse Matrices 10-6

Five-Minute Check (over Lesson 3–7)
CCSS
Then/Now
New Vocabulary
Key Concept: Identity Matrix for Multiplication
Example 1: Verify Inverse Matrices
Key Concept: Inverse of a 2 × 2 Matrix
Example 2: Find the Inverse of a Matrix
Example 3: Real-World Example: Solve a System of
Equations
Over Lesson 3–7
Write a matrix expression you could
use to calculate the area of this plot of
land.
Use Cramer’s Rule to solve the system of equation.
2x + y – z = 5
x + 4y + 2z = 16
15x + 6y – 2z = 12
Over Lesson 3–7
A. 2
B. 7
C. 14
D. 22
Over Lesson 3–7
A. –66
B. –48
C. 20
D. 160
Over Lesson 3–7
Write a matrix expression you could
use to calculate the area of this plot of
land.
A.
B.
C.
D.
Over Lesson 3–7
Use Cramer’s Rule to solve the system of equation.
2x + y – z = 5
x + 4y + 2z = 16
15x + 6y – 2z = 12
A. (2, 3, 0)
B. (4, 2, 2)
C. (–2, 6, –3)
D. (–1, 3, 3)
Content Standards
A.CED.3 Represent constraints by equations
or inequalities, and by systems of equations
and/or inequalities, and interpret solutions as
viable or nonviable options in a modeling
context.
Mathematical Practices
5 Use appropriate tools strategically.
You solved systems of linear equations
algebraically.
• Find the inverse of a 2 × 2 matrix.
• Write and solve matrix equations for a
system of equations.
• identity matrix
• square matrix
• inverse matrix
• matrix equation
• variable matrix
• constant matrix
Why do we care about inverse matrices?
Because inverses “undo”
A •X = B
A-1• A •X = A-1•B
I•X = A-1•B
X = A-1•B
Verify Inverse Matrices
A. Determine whether X and Y are inverses.
If X and Y are inverses, then X ● Y = Y ● X = I.
Answer: Since X ● Y = Y ● X = I, X and Y are inverses.
Verify Inverse Matrices
B. Determine whether P and Q are inverses.
If P and Q are inverses, then P ● Q = Q ● P = I.
Answer: Since P ● Q  I, they are not inverses.
A. Determine whether the matrices are inverses.
A. yes
B. no
C. not enough information
D. sometimes
B. Determine whether the matrices are inverses.
A. yes
B. no
C. not enough information
D. sometimes
𝐴−1
1
𝑑
=
𝑑𝑒𝑡𝐴 −𝑐
−𝑏
𝑎
Find the Inverse of a Matrix
A. Find the inverse of the matrix, if it exists.
det 𝑆 =
−1
8
0
=2 −0=2
−2
Since the determinant is not equal to 0, S –1 exists.
𝑆
−1
1
𝑑
=
𝑑𝑒𝑡𝑆 −𝑐
1 −2
−𝑏
=
𝑎
2 −8
−1
𝑆 −1 =
−4
0
1
−
2
0
−1
Find the Inverse of a Matrix
A. Find the inverse of the matrix, if it exists.
Check If the product is I, then they are inverse.
−1
0
−1 0
1
•
−4 −
−8 −2
2
−1
0
1 • −1
−4 −
−8
2
0+0 = 1 0
0 1
0+1
1+0 0+0
=
4−4 0+1
−1 0
1
=
−4 −
2
0
−2
𝑆 −1
1+0
=
−8 + 8
1 0
=
0 1
Find the Inverse of a Matrix
B. Find the inverse of the matrix, if it exists.
−4
det 𝑇 =
−2
6
= −12 + 12 = 0
3
Since the determinant is equal to 0, T –1 does not exist.
A. Find the inverse of the matrix, if it exists.
A.
B.
C.
D. No inverse
exists.
B. Find the inverse of the matrix, if it exists.
A.
B.
C.
D. No inverse
exists.
Solve a System of Equations
To Solve a System of Equations Using Inverse Matrices:
Let A be the coefficient matrix
X be the variable matrix
B be the constant matrix
Matrix equation:
Then
𝑥
11
2 −5
A •X = B
• 𝑦 =
−13
−3 4
A-1• A •X = A-1•B
X = A-1•B
Example:
2x – 5y = 11
-3x + 4y = -13
Inverse matrix:
1 4 5
−
7 3 2
Solve a System of Equations
To Solve a System of Equations Using Inverse Matrices:
Let A be the coefficient matrix
X be the variable matrix
B be the constant matrix
Then
A •X = B
A-1• A •X = A-1•B
X = A-1•B
1 4
−
7 3
1 4
𝑥
5
2 −5
•
• 𝑦 =−
2
−3 4
7 3
𝑥
1 0
3
• 𝑦 =
0 1
−1
So (3,-1)
11
5
•
−13
2
Solve a System of Equations
RENTAL COSTS The Booster Club for North High
School plans a picnic. The rental company charges
$15 to rent a popcorn machine and $18 to rent a
water cooler. The club spends $261 for a total of
15 items. How many of each do they rent?
System of equations:
x + y = 15
15x + 18y = 261
Matrix equation:
𝑥
1
1
15
• 𝑦 =
15 18
261
Inverse matrix:
1 18 −1
3 −15 1
Solve a System of Equations
RENTAL COSTS The Booster Club for North High
School plans a picnic. The rental company charges
$15 to rent a popcorn machine and $18 to rent a
water cooler. The club spends $261 for a total of
15 items. How many of each do they rent?
System of equations:
x + y = 15
15x + 18y = 261
1 18 −1
1 18 −1
𝑥
1
1
15
•
• 𝑦 =
•
15 18
261
3 −15 1
3 −15 1
𝑥
1 0
3
• 𝑦 =
0 1
12
So (3,12)
Answer: The club rents 3 popcorn machines and
12 water coolers.
Use a matrix equation to solve the system of
equations.
3x + 4y = –10
x – 2y = 10
A. (–2, 4)
B. (2, –4)
C. (–4, 2)
D. no solution