s
Section 5.7 Matrices
465
29. −x + 2y + z = −1
3x − 5y − z = 5
2x − y + 3z = 22
30. x − y + z = 6
x + 2y − 3z = 10
2x + 3y + 5z = 12
31. x + 2y − z = −1
x − 3y + z = 2
2x + y + 2z = 6
32. 2x − 7y + z + 2w = 5
x + y − 2z + 3w = 8
4x + 2y + z − 4w = 12
5x + 3y − z − w = 10
33.
x + 2y − z + w = −3
2x − y + 4z + 2w = 33
x − y + 3z − 7w =
6
−3x + 3y + z + 4w = −12
34. x − 2y + 3z − w = 7
2x + 5y − 7z + 2w = 12
4x − 3y + 12z + w = 8
35. Find an example of a system of 3 linear equations with 2 unknowns that has a solution. Explain what happens when you use Gaussian elimination on this system.
36. Find an example of a system of 4 linear equations with 3 unknowns that has a solution. Explain what happens when you use Gaussian elimination on this system.
37. You purchase an ancient Egyptian medallion at the State Fair from a vendor who swears it is pure gold.
The medallion weighs 859.4 grams and its volume in cubic centimeters is 52. You suspect that the medallion is actually a mixture of copper and gold. You know that copper weighs 9 grams per cubic centimeter
and that gold weighs 19.3 grams per cubic centimeter. Set up and solve a system of equations to find the
percentage of copper by volume in the medallion.
38. Cell phone Plan A charges a flat monthly fee of $30.00 for the first 400 minutes, plus $0.07 for each
minute > 400. Plan B charges a flat monthly fee of $45.00 for the first 600 minutes, plus $0.19 for each
minute > 600. You know you will use more than 600 minutes per month.
a. At what number of minutes do the plans cost the same amount per month, and what is that amount?
b. Above this number of minutes, which plan is more expensive?
39. If a is an n × n invertible matrix, the following method can be used to find a
−1
.
1. Operate on a using any combination of the two following elementary row operations until the resulting
matrix is the n × n identity matrix i.
i. Multiply all the elements in any one row of a by a non-zero scalar.
ii. Add a scalar multiple of any row to any other row.
2. At the same time, perform exactly the same sequence of operations on the n × n identity matrix i.
3. The matrix that results from i after step 2 is a
−1
.
Use this method to find the inverse of matrix a in Exercise 24(a).
Section 5.7 Matrices
465
29. −x + 2y + z = −1
3x − 5y − z = 5
2x − y + 3z = 22
30. x − y + z = 6
x + 2y − 3z = 10
2x + 3y + 5z = 12
31. x + 2y − z = −1
x − 3y + z = 2
2x + y + 2z = 6
32. 2x − 7y + z + 2w = 5
x + y − 2z + 3w = 8
4x + 2y + z − 4w = 12
5x + 3y − z − w = 10
33.
x + 2y − z + w = −3
2x − y + 4z + 2w = 33
x − y + 3z − 7w =
6
−3x + 3y + z + 4w = −12
34. x − 2y + 3z − w = 7
2x + 5y − 7z + 2w = 12
4x − 3y + 12z + w = 8
35. Find an example of a system of 3 linear equations with 2 unknowns that has a solution. Explain what happens when you use Gaussian elimination on this system.
36. Find an example of a system of 4 linear equations with 3 unknowns that has a solution. Explain what happens when you use Gaussian elimination on this system.
37. You purchase an ancient Egyptian medallion at the State Fair from a vendor who swears it is pure gold.
The medallion weighs 859.4 grams and its volume in cubic centimeters is 52. You suspect that the medallion is actually a mixture of copper and gold. You know that copper weighs 9 grams per cubic centimeter
and that gold weighs 19.3 grams per cubic centimeter. Set up and solve a system of equations to find the
percentage of copper by volume in the medallion.
38. Cell phone Plan A charges a flat monthly fee of $30.00 for the first 400 minutes, plus $0.07 for each
minute > 400. Plan B charges a flat monthly fee of $45.00 for the first 600 minutes, plus $0.19 for each
minute > 600. You know you will use more than 600 minutes per month.
a. At what number of minutes do the plans cost the same amount per month, and what is that amount?
b. Above this number of minutes, which plan is more expensive?
39. If a is an n × n invertible matrix, the following method can be used to find a
−1
.
1. Operate on a using any combination of the two following elementary row operations until the resulting
matrix is the n × n identity matrix i.
i. Multiply all the elements in any one row of a by a non-zero scalar.
ii. Add a scalar multiple of any row to any other row.
2. At the same time, perform exactly the same sequence of operations on the n × n identity matrix i.
3. The matrix that results from i after step 2 is a
−1
.
Use this method to find the inverse of matrix a in Exercise 24(a).