EXAMPLE 1 Evaluate determinants Evaluate the determinant of the matrix. a.54 31 SOLUTION b.2 3 4 1 1...
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Transcript of EXAMPLE 1 Evaluate determinants Evaluate the determinant of the matrix. a.54 31 SOLUTION b.2 3 4 1 1...
EXAMPLE 1 Evaluate determinants
Evaluate the determinant of the matrix.
a. 5 43 1
SOLUTION
b. 2
34
114
3
20
– –
– –
EXAMPLE 2 Find the area of a triangular region
Sea LionsOff the coast of California lies a triangular region of the Pacific Ocean where huge populations of sea lions and seals live. The triangle is formed by imaginary lines connecting Bodega Bay, the Farallon Islands, and Año Nuevo Island, as shown. (In the map, the coordinates are measured in miles.) Use a determinant to estimate the area of the region.
EXAMPLE 2 Find the area of a triangular region
SOLUTIONThe approximate coordinates of the vertices of the triangular region are ( 1, 41), (38, 43), and (0, 0). So, the area of the region is:
– –
Area =1
038
4143
1
11
– –
0+ –
12
–12
+= [(43 + 0 + 0) (0 + 0 + 1558)] –
= 757.5
The area of the region is about 758 square miles.
1
038
4143
1
11
– –
0+ 1
2 –1
0
4143 –0
38 –
=
EXAMPLE 3
Use Cramer’s rule to solve this system:3x 5y = 219x + 4y = 6
– – –
SOLUTION
STEP 1Evaluate the determinant of the coefficient matrix.
9 43 5 –
Use Cramer’s rule for a 2 2 system
–= 45 12 = 57 – –
EXAMPLE 3
STEP 2Apply Cramer’s rule because the determinant is not 0.
y =
9 63 21 – –
–
57 – = 57 – 189 ( 18) – –
= 57 –171
= 3 –
ANSWER
The solution is ( 2, 3). –
Use Cramer’s rule for a 2 2 system
x =
6 421 5 – –
–
57 – = 57 – 30 ( 84) – –
= 57 –114
= 2 –
EXAMPLE 3
CHECK
Check this solution in the original equations.
– 21= – 21
Use Cramer’s rule for a 2 2 system
9x + 4y = 6 –9( 2) + 4(3) = 6 – – ?
18 + 12 = 6 – ? – – 6= – 6
?3( 2) 5(3) = 21 – ––3x 5y = 21 ––
6 15 = 21 – ? – –
EXAMPLE 4 Solve a multi-step problem
CHEMISTRY
The atomic weights of three compounds are shown. Use a linear system and Cramer’s rule to find the atomic weights of carbon (C), hydrogen (H), and oxygen (O).
EXAMPLE 4 Solve a multi-step problem
SOLUTION
Write a linear system using the formula for each compound. Let C, H, and O represent the atomic weights of carbon, hydrogen, and oxygen.
6C + 12H + 6O = 180 C + 2O = 44
2H + 2O = 34
STEP 1
EXAMPLE 4 Solve a multi-step problem
STEP 2Evaluate the determinant of the coefficient matrix.
6
01
120
6
22
2
6
01
1202
= (0 + 0 + 12) (0 + 24 + 24) = 36– –
STEP 3Apply Cramer’s rule because the determinant is not 0.
180
34
120
6
22
2
6
01
180 44
6
22
34
6
01
120
180
3444
2C = H = O =
44
36 36 36– – –
EXAMPLE 4 Solve a multi-step problem
576 36–
–=
3636–
–=
43236–
–=
= 12 = 1 = 16
ANSWER
The atomic weights of carbon, hydrogen, and oxygen are 12, 1, and 16, respectively.
EXAMPLE 1 Find the inverse of a 2 × 2 matrix
A–1 = 115 – 16
5 – 8
– 2 3
Find the inverse of A = .
3 8
2 5
= – 1 =– 5 8
2 – 3
5 – 8
– 2 3
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 = 18 – 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=
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.
EXAMPLE 3 Find the inverse of a 3 × 3 matrix
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) variables (X) 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 = 18 – (–3)
4111
11
3112
11–
STEP 3 Multiply the matrix of constants by A–1 on the left.
X = A–1B =
4111
11
311
11– 2
19
– 7=
5
– 3=
x
y
EXAMPLE 4 Solve a linear system
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.