HW: Pg. 219 #16-26e, 31, 33
Jan 12, 2016
HW: Pg. 219 #16-26e, 31, 33
HW: Pg. 219-220 #37, 41, 45, 49, 59
HW: Pg. 219-220 #37, 41, 45, 49, 59
HW: Pg. 219-220 #37, 41, 45, 49, 59
HW: Quiz 1 Pg. 221 #7-17o
HW: Quiz 1 Pg. 221 #7-17o
4.4 Identity and Inverse Matrices
EXAMPLE 1 Find the inverse of a 2 X 2 matrix
Find the inverse of A = .3 8
2 5
1
1
8
2
-8 -8 -8
-2 -2 -2
5
5
3
3 3 3
5 5
5 5
5
3 3
3
1 1 1
3(15)-2(8) 15 16 1
+1
-8 8
-2 2
+
8
2
-
+
+ -
A
A
A
so
GUIDED PRACTICE for Example 1
Find the inverse of the matrix.
1. 6 1
2 4
2
11
1
11
3
11–
1
22–
ANSWER
2. –1 5
–4 8
2
3
1
3
5
12–
1
12–
ANSWER
3. –3 –4
–1 –2
–1
1
2
2
3
2
–
ANSWER
EXAMPLE 2 Solve a matrix equation
SOLUTION
Begin by finding the inverse of A.
4 7
1 2=
Solve the matrix equation AX = B for the 2 × 2 matrix X.
2 –7
–1 4
–21 3
12 –2
A B
X =
A–1 =1
8 – 7
4 7
1 2
EXAMPLE 2 Solve a matrix equation
To solve the equation for X, multiply both sides of the equation by A– 1 on the left.
A–1 AX = A–1 B
IX = A–1 B
X = A–1 BX =0 –2
3 –1
4 7
1 2
–21 3
12 –2=
2 –7
–1 4
4 7
1 2X
X 1 0
0 1
0 –2
3 –1=
GUIDED PRACTICE for Example 2
4. Solve the matrix equation –4 1
0 6X =
8 9
24 6
–1 –2
4 1
ANSWER
EXAMPLE 3 Find the inverse of a 3 × 3 matrix
Use a graphing calculator to find the inverse of A. Then use the calculator to verify your result.
2 1 – 2
5 3 0
4 3 8
A =
SOLUTION
Enter matrix A into a graphing calculator and calculate A–1. Then compute AA–1and A–1A to verify that you obtain the 3 × 3 identity matrix.
GUIDED PRACTICE for Example 3
5. 2 –2 0
2 0 –2
12 –4 –6A =
Use a graphing calculator to find the inverse of the matrix A. Check the result by showing that AA-1= I and A-1A = I.
GUIDED PRACTICE for Example 3
6. – 3 4 5
1 5 0
5 2 2
A =
7. 2 1 – 2
5 3 0
4 3 8
A =
EXAMPLE 4 Solve a linear system
Use an inverse matrix to solve the linear system.
2x – 3y = 19
x + 4y = –7
Equation 1
Equation 2
SOLUTION
STEP 1 Write the linear system as a matrix equation AX = B.
coefficient matrix of matrix of
matrix (A) (X) variables constants(B)
2 –3
1 4
. x
y
19
–7=
EXAMPLE 4 Solve a linear system
STEP 2 Find the inverse of matrix A.
4 3
–1 2
=A–1 =1
8 – (–3)
4
11
1
11
3
11
2
11–
STEP 3 Multiply the matrix of constants by A–1 on the left.
X = A–1B =
4
11
1
11
3
11
11– 2
19
–7=
5
–3=
x
y
The solution of the system is (5, – 3).
ANSWER
CHECK 2(5) – 3(–3) = 10 + 9 = 19
5 + 4(–3) = 5 – 12 = –7
EXAMPLE 5 Solve a multi-step problem
Gifts
A company sells three types of movie gift baskets. A basic basket with 2 movie passes and 1 package of microwave popcorn costs $15.50. A medium basket with 2 movie passes, 2 packages of popcorn, and 1 DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVDs costs $72.50. Find the cost of each item in the gift baskets.
EXAMPLE 5 Solve a multi-step problem
SOLUTION
STEP 1 Write verbal models for the situation.
EXAMPLE 5 Solve a multi-step problem
STEP 2 Write a system of equations. Let m be the cost of a movie pass, p be the cost of a package of popcorn, and d be the cost of a DVD.
2m + p = 15.50 Equation 1
2m + 2p + d = 37.00 Equation 2
4m + 3p + 2d = 72.50 Equation 3
STEP 3 Rewrite the system as a matrix equation.
2 1 0
2 2 1
4 3 2
m
p
d
15.50
37.00
72.50
=
EXAMPLE 5 Solve a multi-step problem
STEP 4 Enter the coefficient matrix A and the matrix of constants B into a graphing calculator. Then find the solution X = A–1B.
A movie pass costs $7, a package of popcorn costs $1.50, and a DVD costs $20.
GUIDED PRACTICE for Examples 4 and 5
Use an inverse matrix to solve the linear system.
4x + y = 10
3x + 5y = –1
8.
(3, –2)
ANSWER
9. 2x – y = – 6
6x – 3y = – 18
infinitely many solutions
ANSWER
10. 3x – y = –5
–4x + 2y = 8
(– 1, 2)
ANSWER
11. What if? In Example 5, how does the answer change if a basic basket costs $17, a medium basket costs $35, and a super basket costs $69?
movie pass: $8package of popcorn: $1
DVD: $17
ANSWER
Homework:
Pg. 227 #17-23o, 28, 30, 31, 33-36