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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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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
In each of these procedures, the numbers (coefficients) are what ischanged–not the variables.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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
3x + 4y+2x = 5
x + 2y+0z = 4
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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
3x + 4y+2x = 5
x + 2y+0z = 4
A =
2 3 73 4 21 2 0
Notice how each row corresponds to an equation and each columnto a single variable.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
2x + 3y+7z = 1
3x + 4y+2x = 5
x + 2y+0z = 4
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
2x + 3y+7z = 1
3x + 4y+2x = 5
x + 2y+0z = 4
2 3 7 13 4 2 51 2 0 4
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
2x + 4y= 8
x − y = 1
1 Interchanging rows
2 Multiplying a row by a non-zero constant
3 Adding a multiple of one row to another
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
2x + 4y= 8
x − y = 1
1 Interchanging rows
2 Multiplying a row by a non-zero constant
3 Adding a multiple of one row to another
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
2x + 4y= 8
x − y = 1
1 Interchanging rowsSwap row 1 and row 2.
2 Multiplying a row by a non-zero constant
3 Adding a multiple of one row to another
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
2x + 4y= 8
x − y = 1
1 Interchanging rowsSwap row 1 and row 2.
2 Multiplying a row by a non-zero constant
3 Adding a multiple of one row to another
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
2x + 4y= 8
x − y = 1
1 Interchanging rowsSwap row 1 and row 2.
2 Multiplying a row by a non-zero constantMultiply row 1 by 1
2 .
3 Adding a multiple of one row to another
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
2x + 4y= 8
x − y = 1
1 Interchanging rowsSwap row 1 and row 2.
2 Multiplying a row by a non-zero constantMultiply row 1 by 1
2 .
3 Adding a multiple of one row to another
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
2x + 4y= 8
x − y = 1
1 Interchanging rowsSwap row 1 and row 2.
2 Multiplying a row by a non-zero constantMultiply row 1 by 1
2 .
3 Adding a multiple of one row to anotherAdd 1
2 row 1 to row 2.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Identifying Row Echelon Form
Row Echelon Form
Which of the matrices below are in row echelon from?
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Identifying Row Echelon Form
Row Echelon Form
Which of the matrices below are in row echelon from?1 2 0 70 0 1 −10 1 0 0
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Identifying Row Echelon Form
Row Echelon Form
Which of the matrices below are in row echelon from?1 2 0 70 0 1 −10 1 0 0
1 −2 7 −120 1 2 70 0 1 4
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Identifying Row Echelon Form
Row Echelon Form
Which of the matrices below are in row echelon from?1 2 0 70 0 1 −10 1 0 0
1 −2 7 −120 1 2 70 0 1 4
1 5 3 −10 2 4 −20 0 0 6
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Identifying Row Echelon Form
Row Echelon Form
Which of the matrices below are in row echelon from?1 2 0 70 0 1 −10 1 0 0
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
z = 4
Which can be solved easily bysubstitution (plug z = 4 intoy + 2z = 7 and so on).
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Solving with Augmented Matrices
Solve
Solve the system of equations below using an augmented matrixand row operations.
2x+4y+2z =−4
3x+6y+4z =−5
y+2z = 1
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Solving with Augmented Matrices
Solve
Solve the system of equations below using an augmented matrixand row operations.
2x+4y+2z =−4
3x+6y+4z =−5
y+2z = 1
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Solving with Augmented Matrices
Solve
Solve the system of equations below using an augmented matrixand row operations.
2x+4y+2z =−4
3x+6y+4z =−5
y+2z = 1
2 4 2 −43 6 4 −50 1 2 1
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Solving with Augmented Matrices
Solve
Solve the system of equations below using an augmented matrixand row operations.
2x+4y+2z =−4
3x+6y+4z =−5
y+2z = 1
2 4 2 −43 6 4 −50 1 2 1
⇒
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Solving with Augmented Matrices
Solve
Solve the system of equations below using an augmented matrixand row operations.
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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
Solving with Augmented Matrices
Solve
Solve the system of equations below using an augmented matrixand row operations.
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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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).
Solve
x+y =−2
x+y =−5
Solve
x+ y =−2
3x+3y =−6
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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).
Solve
x+y =−2
x+y =−5
Solve
x+ y =−2
3x+3y =−6
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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).
Solve
x+y =−2
x+y =−5
Row operations yield:[1 1 −20 0 −3
]This system is inconsistent.
Solve
x+ y =−2
3x+3y =−6
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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).
Solve
x+y =−2
x+y =−5
Row operations yield:[1 1 −20 0 −3
]This system is inconsistent.
Solve
x+ y =−2
3x+3y =−6
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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).
Solve
x+y =−2
x+y =−5
Row operations yield:[1 1 −20 0 −3
]This system is inconsistent.
Solve
x+ y =−2
3x+3y =−6
Row operations yield:[1 1 −20 0 0
]This system is dependent.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix 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.
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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!
From a Set of Equations to a Matrix The Augmented Matrix and Row Operations Solving Systems of Equations with an Augmented Matrix Conclusion
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!
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