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Solve systems of equations using matrices.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
9.3 Matrices and Systems of Equations
Matrices
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
The system
can be expressed as
where we have omitted the variables and replaced the equals signs with a vertical line.
4 2 3
1 5 4
4 2 3
5 4
x y
x y
Matrices
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
A rectangular array of numbers such as
is called a matrix (plural, matrices). The matrix is an augmented matrix because it contains not only the coefficients but also the constant terms.
The matrix is called the coefficient matrix.
4 2 3
1 5 4
4 2
1 5
Matrices continued
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The rows of a matrix are horizontal.
The columns of a matrix are vertical.
The matrix shown has 2 rows and 3 columns.
A matrix with m rows and n columns is said to be of order m n.
When m = n the matrix is said to be square.
1 2 3
4 5 6
Gaussian Elimination with Matrices
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Row-Equivalent Operations
1.Interchange any two rows.
2.Multiply each entry in a row by the same nonzero constant.
3.Add a nonzero multiple of one row to another row.
We can use the operations above on an augmented
matrix to solve the system.
Example
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Solve the following system:
First, we write the augmented matrix, writing 0 for the missing y-term in the last equation.
Our goal is to find a reduced row-echelon form matrix.Each column contains a 1 on the diagonal and has 0’s everywhere else.
2 8
3 2 1
4 5 23
x y z
x y z
x z
2 1 1 8
1 3 2 1
4 0 5 23
Example
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Which of the following matrices are in reduced row-echelon form?
a) b)
c) d)
1 6 7 5
0 1 3 4
0 0 1 8
0 2 4 1
0 0 0 0
1 2 7 6
0 0 0 0
0 1 4 2
1 0 0 3.5
0 1 0 0.7
0 0 1 4.5
Gauss-Jordan Elimination
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We perform row-equivalent operations on a matrix to obtain a until we have a matrix in reduced row-echelon form.
Example: Find the solution.
0
0 0
3 2 1
1 2
3
1
1
1
Special Systems
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When a row consists entirely of 0’s, the equations are dependent. For example, in the matrix
the system is equivalent to
4 6,
4 8.
x z
y z
1 0 4 6
0 1 4 8
0 0 0 0
Special Systems
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
When we obtain a row whose only nonzero entry occurs in the last column, we have an inconsistent system of equations. For example, in the matrix
the last row corresponds to the false equation 0 = 9, so we know the original system has no solution.
1 0 4 6
0 1 4 8
0 0 0 9
Practice
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Solve the system of given systems of equations.
