Mat 150 – Class #22

Objectives

Represent systems of equations with matricesFind dimensions of matrices Identify square matrices Identify an identity matrixForm an augmented matrix Identify a coefficient matrixReduce a matrix with row operationsReduce a matrix to its row-echelon formSolve systems of equations using the Gauss-Jordan elimination method

Matrix Representation of Systems of Equations

When given a system of equations, it can be written as a matrix.

2 3 1

2 3

3 9

x y z

x y z

x y z

2 3 1 3

1 1 2 3

3 1 1 9

The column to the right of the vertical line, containing the constants of the equations, is called the augment of the matrix, and a matrix containing an augment is called an augmented matrix.

A square matrix that has 1’s down its diagonal and 0’s everywhere else, like matrix I below, is called an identity matrix.

Any augmented matrix that has 1’s or 0’s on the diagonal of its coefficient part and 0's below the diagonal is said to be in row-echelon form.

Example

Solve the system

SolutionBegin by writing the augmented matrix.

2 3 1

2 3

3 9

x y z

x y z

x y z

2 3 1 3

1 1 2 3

3 1 1 9

Example (cont)

Interchange equations 1 and 2; thus change rows 1 and 2.

Get 0 as the first entry in the second row and the first entry of the third row.–2R1 + R2 →R2

–3R1 + R3 →R3

2 3

2 3 1

3 9

x y z

x y z

x y z

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

2 3 1 1

3 1 1 9

1 1 2 3

0 1 3 5

0 4 7 18

Example (cont)

2 3

3 5

4 7 18

x y z

y z

y z

2 3

3 5

19 38

x y z

y z

z

2 3

3 5

2

x y z

y z

z

–1R2 → R2

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

0 1 3 5

0 0 19 38

–4R2 + R3→ R3

(–1/19)R3→ R3

1 1 2 3

0 1 3 5

0 0 1 2

Example (cont)

The matrix is now in row-echlon form. The equivalent system can be solved by back substitution.

The solution is (2, 1, 2).

Gauss-Jordan Elimination

The augmented matrix representing n equations in n variables is said to be in reduced row-echelon form if it has 1’s or 0’s on the diagonal of its coefficient partand 0’s everywhere else.

2 3

3 5

2

x y z

y z

z

1 1 2 3

0 1 3 5

0 0 1 2

Example

Solve the system

SolutionRepresent by the augmented matrix.

3

2 4 5

0

2

x y z w

x y z w

x z w

y z w

1 1 1 1 3

1 2 1 4 5

1 0 1 1 0

0 1 1 1 2

Example (cont)

We can enter this augmented matrix into a graphing calculator and reduce the matrix to row-echelon form.

x = 1, y = 11, z = –4, w = –5, or (1, 11, –4, –5)

Dependent and Inconsistent Systems

A system with fewer equations than variables has either infinitely many solutions or no solutions.

If a row of the reduced row-echelon coefficient matrix associated with a system contains all 0’s and the augment of that row contains a nonzero number, the system has no solution and is an inconsistent system.

If a row of the reduced 3 × 3 row-echelon coefficient matrix associated with a system contains all 0’s and the augment of that row also contains 0, then there areinfinitely many solutions and is a dependent system.

Example

Ace Trucking Company has an order for delivery of three products: A, B, and C. If the company can carry 30,000 cubic feet and 62,000 pounds and is insured for $276,000, how many units of each product canbe carried?

Example (cont)

If we represent the number of units of product A by x, the number of units of product B by y, and the number of units of product C by z, then we can write a system of equations to represent the problem.

25 22 30 30,000 Volume

25 + 38 70 62,000 Weight

150 180 300 276,000 Value

x y z

x y z

x y z

Example (cont)

The Gauss-Jordan elimination method gives

25 22 30 30,000

25 38 70 62,000

150 180 300 276,000

Example (cont)

To save time, if we use a graphing calculator.

The solution to this system is x = –560 + z, y = 2000 – 2.5z, with the values of z limited so that all values are nonnegative integers. Product C: 560 ≤ z ≤ 800 (z is an even integer)Product B: y = 2000 – 2.5zProduct A: x = –560 + z

Assignment

Pg. 518-521#15-21 (Must show work)#23-31 (May use the calculator) #34, #39 and #42