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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