math 105: finite mathematics 2-2: augmented matrices

From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion

MATH 105: Finite Mathematics2-2: Augmented Matrices

Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

Outline

1 From a Set of Equations to a Matrix

2 The Augmented Matrix and Row Operations

3 Solving Systems of Equations with an Augmented Matrix

4 Conclusion

Outline

4 Conclusion

The Elimination Process

In the last section we solved systems of equations by elimination.There were several things that we could do to equations toeliminate a variable.

Elimination Tools

In solving a system of equations by elimination, the tools whichone can use are:

Switch the order in which equations are written

Add two equations together

Multiply any equation by a constant

Elimination Tools

In each of these procedures, the numbers (coefficients) are what ischanged–not the variables.

Getting Rid of Variables

To help make the process easier, we can concentrate just on thenumbers and drop the variables completely.

Finding a Matrix of Coefficients

Find the matrix of coefficients for the following system ofequations.

2x + 3y+7z = 1

4y + 2z+3x = 5

2y + x = 4

2x + 3y+7z = 1

4y + 2z+3x = 5

2y + x = 4

2x + 3y+7z = 1

3x + 4y+2x = 5

x + 2y+0z = 4

2x + 3y+7z = 1

4y + 2z+3x = 5

2y + x = 4

2x + 3y+7z = 1

3x + 4y+2x = 5

x + 2y+0z = 4

2 3 73 4 21 2 0

Notice how each row corresponds to an equation and each columnto a single variable.

The Augmented Matrix

The matrix of coefficients, usually called A, involves only the coefficientson the left-hand side of the equations. The constants on the right hadside can be added to this to produce an augmented matrix.

Augmented Matrix

The augmented matrix for a system of equations is a matrix composed ofthe matrix of coefficients, A, together with the constant matrix, B. It iswritten as [

A | B]

Finding an Augmented Matrix

Find the augmented matrix for the system of equations seen before.

Augmented Matrix

A | B]

Augmented Matrix

A | B]

Augmented Matrix

A | B]

2x + 3y+7z = 1

3x + 4y+2x = 5

x + 2y+0z = 4

Augmented Matrix

A | B]

2x + 3y+7z = 1

3x + 4y+2x = 5

x + 2y+0z = 4

2 3 7 13 4 2 51 2 0 4

Outline

4 Conclusion

Row Operations

Since each row of this new matrix represents an equation, we cando the same things to rows that we used to solve systems ofequations by elimination.

Allowed Row Operations

The system of equations represented by a given augmented matrixwill have the same solutions as that represented by a newaugmented matrix produced by:

Interchanging rows

Multiplying a row by a non-zero constant

Replacing a row with the sum of that row and a constantmultiple of some other row.

Row Operations

Interchanging rows

Row Operations

Interchanging rows

Row Operations

Interchanging rows

Row Operations

Interchanging rows

Verifying Row Operations Work

Example

Verify that each of the three operations produce an augmentedmatrix representing a system of equations with the same solutionas the original system of equations.

1 Interchanging rows

2 Multiplying a row by a non-zero constant

3 Adding a multiple of one row to another

Example

2x + 4y= 8

x − y = 1

Example

2x + 4y= 8

x − y = 1

Example

2x + 4y= 8

x − y = 1

1 Interchanging rowsSwap row 1 and row 2.

Example

2x + 4y= 8

x − y = 1

Example

2x + 4y= 8

x − y = 1

2 Multiplying a row by a non-zero constantMultiply row 1 by 1

Example

2x + 4y= 8

x − y = 1

Example

2x + 4y= 8

x − y = 1

3 Adding a multiple of one row to anotherAdd 1

2 row 1 to row 2.

Row Echelon Form

So if row operations lead to systems with the same solution, ourgoal is to produce a system which is easy to solve.

Row Echelon Form

A matrix is in row echelon from when:

1 The entry in row 1, column 1 is a 1, and 0s appear below it(this is called a leading 1)

2 The first non-zero entry of each row is a leading 1 with 0sbelow it, and it appears to the right of the leading one in anyrow above.

3 Any rows that contain all 0s to the left of the vertical barappear at the bottom.

Row Echelon Form

Identifying Row Echelon Form

Row Echelon Form

Which of the matrices below are in row echelon from?

Row Echelon Form

Which of the matrices below are in row echelon from?1 2 0 70 0 1 −10 1 0 0

Row Echelon Form

1 −2 7 −120 1 2 70 0 1 4

Row Echelon Form

1 −2 7 −120 1 2 70 0 1 4

1 5 3 −10 2 4 −20 0 0 6

Row Echelon Form

1 −2 7 −120 1 2 70 0 1 4

1 5 3 −10 2 4 −20 0 0 6

Row echelon form is nice because if you translate an augmentedmatrix like the middle one above back into an equation, you get

x − 2y+7z = −1

y +2z = 7

Which can be solved easily bysubstitution (plug z = 4 intoy + 2z = 7 and so on).

Outline

4 Conclusion

Solving with Augmented Matrices

Solve the system of equations below using an augmented matrixand row operations.

2x+4y+2z =−4

3x+6y+4z =−5

y+2z = 1

2x+4y+2z =−4

3x+6y+4z =−5

y+2z = 1

2x+4y+2z =−4

3x+6y+4z =−5

y+2z = 1

2 4 2 −43 6 4 −50 1 2 1

2x+4y+2z =−4

3x+6y+4z =−5

y+2z = 1

2 4 2 −43 6 4 −50 1 2 1

2x+4y+2z =−4

3x+6y+4z =−5

y+2z = 1

2 4 2 −43 6 4 −50 1 2 1

1 2 1 −20 1 2 10 0 1 1

2x+4y+2z =−4

3x+6y+4z =−5

y+2z = 1

2 4 2 −43 6 4 −50 1 2 1

1 2 1 −20 1 2 10 0 1 1

x = −1, y = −1, z = 1

Inconsistent and Dependent Solutions

As we saw in the last section, not all systems of equations willhave a unique solution. Using augmented matrices, one canidentify systems which are inconsistent (no solution) or dependent(infinitely many solutions).

x+y =−2

x+y =−5

x+ y =−2

3x+3y =−6

x+y =−2

x+y =−5

x+ y =−2

3x+3y =−6

x+y =−2

x+y =−5

Row operations yield:[1 1 −20 0 −3

]This system is inconsistent.

x+ y =−2

3x+3y =−6

x+y =−2

x+y =−5

x+ y =−2

3x+3y =−6

x+y =−2

x+y =−5

x+ y =−2

3x+3y =−6

Row operations yield:[1 1 −20 0 0

]This system is dependent.

Outline

4 Conclusion

Important Concepts

Things to Remember from Section 2-2

1 Writing Augmented Matrices from a System of Equations

2 Performing Row Operations

3 Identifying Row Echelon Form

4 Solving a system of equations using row operations.

Important Concepts

Next Time. . .

In section 2-3 we will work with systems of equations in which thenumber of variables is not the same as the number of equations.

For next time

Read section 2-3

Play in the snow!

Next Time. . .

In section 2-3 we will work with systems of equations in which thenumber of variables is not the same as the number of equations.

For next time

Read section 2-3

Play in the snow!

math 105: finite mathematics 2-2: augmented matrices

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