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Solving Systems of Linear Equations using Augmented Matrices, Study notes of Mathematics

How to use augmented matrices to solve systems of linear equations through various elementary row operations. It covers interchanging equations, multiplying equations by nonzero constants, and adding constant multiples of one equation to another. The document also discusses the concept of matrices, including their size, identity matrices, and position of elements. The document concludes by forming an augmented matrix, solving it using elementary row operations, and finding the solution.

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

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Augmented Matrices - page 1
Using Augmented Matrices to Solve Systems of Linear Equations
1. Elementary Row Operations
To solve the linear system algebraically, these steps could be used.
x5yz11
3z12
2x4y2z8
+=−
=
+−=
All of the following operations yield a system which is equivalent to the original. (Equivalent
systems have the same solution.)
Interchange equations 2 and 3
x5yz11
2x4y2z8
3z12
+=−
+−=
=
Multiply equation 3 by
1
3
x5yz11
2x4y2z8
z1
+=−
+−=
=
Multiply equation 2 by 1
2
x5yz11
x2yz4
z1
+=−
+=−
=
Add equation 1 to 2 and replace
x5yz11
3y15
z4
+=−
=−
=
equation 2 with the result
Multiply equation 2 by 1
3
x5yz11
y5
z4
+=−
=−
=
Multiply equation 2 by and add it
5
xz14
y5
z4
−=
=−
=
to equation 1; replace equation 1 with
the result
Add equation 3 to equation 1; replace
x18
y5
z4
=
=−
=
equation 1 with the result
The solution is (18,5,4).
pf3

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Using Augmented Matrices to Solve Systems of Linear Equations

Elementary Row Operations

To solve the linear system algebraically, these steps could be used.

x 5y z 11

3z 12

2x 4y 2z 8

All of the following operations yield a system which is equivalent to the original. (Equivalent

systems have the same solution.)

Interchange equations 2 and 3

x 5y z 11

2x 4y 2z 8

3z 12

Multiply equation 3 by

x 5y z 11

2x 4y 2z 8

z 1

Multiply equation 2 by

x 5y z 11

x 2y z 4

z 1

Add equation 1 to 2 and replace

x 5y z 11

3y 15

z 4

equation 2 with the result

Multiply equation 2 by

x 5y z 11

y 5

z 4

Multiply equation 2 by and add it 5

x z 14

y 5

z 4

to equation 1; replace equation 1 with

the result

Add equation 3 to equation 1; replace

x 18

y 5

z 4

equation 1 with the result

The solution is (18, −5, 4).

  1. Operations that Produce Equivalent Systems

a) Two equations are interchanged.

b) An equation is multiplied by a nonzero constant.

c) A constant multiple of one equation is added to another equation.

Matrices

A matrix is a rectangular array of numbers written within brackets. The size of a matrix is always given

in terms of its number of rows and number of columns (in that order!). A 2 x 4 matrix has 2 rows

and 4 columns. Square matrices have the same number of rows and columns. A matrix with a single

column is called a column matrix, and a matrix with a single row is called a row matrix. A square

matrix with all elements on the main diagonal equal to 1 and all other elements equal to 0 is called an

identity matrix. The 3x3 identity matrix is.

The position of an element within a matrix is given by the row and column (in that order!) containing

the element. The element is in row 3 and column 4. 34

a

  1. Elementary Row Operations that Produce Row-Equivalent Matrices

a) Two rows are interchanged i j

R ↔R

b) A row is multiplied by a nonzero constant i i

kR →R

c) A constant multiple of one row is added to another row j i i

kR + R →R

(NOTE : →means "replaces")

Forming an Augmented Matrix

An augmented matrix is associated with each linear system like

x 5y z 11

3z 12

2x 4y 2z 8

The matrix to the left of the bar is called the coefficient matrix.

  1. Solving an Augmented Matrix

To solve a system using an augmented matrix, we must use elementary row operations to change

the coefficient matrix to an identity matrix.

Form the augmented matrix

Interchange rows 2 and 3

2 3

R ↔R