Transcript 3.8 powerpoint

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
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
Verify Inverse Matrices
A. Determine whether X and Y are inverses.
If X and Y are inverses, then X ● Y = Y ● X = I.
Write an equation.
Matrix
multiplication
Verify Inverse Matrices
Write an equation.
Matrix
multiplication
Answer:
Verify Inverse Matrices
Write an equation.
Matrix
multiplication
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.
Write an equation.
Matrix
multiplication
Answer:
Verify Inverse Matrices
B. Determine whether P and Q are inverses.
If P and Q are inverses, then P ● Q = Q ● P = I.
Write an equation.
Matrix
multiplication
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
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
B. Determine whether the matrices are inverses.
A. yes
B. no
C. not enough information
D. sometimes
Find the Inverse of a Matrix
A. Find the inverse of the matrix, if it exists.
Find the determinant.
Since the determinant is not equal to 0, S –1 exists.
Find the Inverse of a Matrix
Definition of inverse
a = –1, b = 0,
c = 8, d = –2
Simplify.
Answer:
Find the Inverse of a Matrix
Definition of inverse
a = –1, b = 0,
c = 8, d = –2
Simplify.
Answer:
Find the Inverse of a Matrix
Check Find the product of the matrices. If the product
is I, then they are inverse.

Find the Inverse of a Matrix
B. Find the inverse of the matrix, if it exists.
Find the value of the determinant.
Answer:
Find the Inverse of a Matrix
B. Find the inverse of the matrix, if it exists.
Find the value of the determinant.
Answer: Since the determinant equals 0, T –1 does not
exist.
A. Find the inverse of the matrix, if it exists.
A.
B.
C.
D. No inverse
exists.
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.
B. Find the inverse of the matrix, if it exists.
A.
B.
C.
D. No inverse
exists.
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?
A system of equations to represent the situation is as
follows.
x + y = 15
15x + 18y = 261
Solve a System of Equations
STEP 1 Find the inverse of the coefficient matrix.
STEP 2 Multiply each side of the matrix equation by
the inverse matrix.
Solve a System of Equations
The solution is (3, 12), where x represents the number of
popcorn machines and y represents the number of water
coolers.
Answer:
Solve a System of Equations
The solution is (3, 12), where x represents the number of
popcorn machines and y represents the number of water
coolers.
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
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