Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
MATH 105: Finite Mathematics2-6: The Inverse of a Matrix
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Outline
1 Solving a Matrix Equation
2 The Inverse of a Matrix
3 Solving Systems of Equations
4 Conclusion
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Outline
1 Solving a Matrix Equation
2 The Inverse of a Matrix
3 Solving Systems of Equations
4 Conclusion
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving Equations
Recall that last time we saw that a system of equations can berepresented as a matrix equation as shown below.
Example
Write the following system of equations in matrix form.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving Equations
Recall that last time we saw that a system of equations can berepresented as a matrix equation as shown below.
Example
Write the following system of equations in matrix form.
{2x + 3y = 73x − 4y = 2
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving Equations
Recall that last time we saw that a system of equations can berepresented as a matrix equation as shown below.
Example
Write the following system of equations in matrix form.
{2x + 3y = 73x − 4y = 2
[2 33 −4
] [xy
]=
[72
]
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving Equations
Recall that last time we saw that a system of equations can berepresented as a matrix equation as shown below.
Example
Write the following system of equations in matrix form.
{2x + 3y = 73x − 4y = 2
[2 33 −4
] [xy
]=
[72
]AX = B
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving Equations
Recall that last time we saw that a system of equations can berepresented as a matrix equation as shown below.
Example
Write the following system of equations in matrix form.
{2x + 3y = 73x − 4y = 2
[2 33 −4
] [xy
]=
[72
]AX = B
If we wish to use the matrix equation on the right to solve a systemof equations, then we need to review how we solve basic equationsinvolving numbers.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving a Simple Equation
The most basic algebra equation, ax = b, is solved using themultiplicative inverse of a.
Example
Solve the equation 3x = 6 for x .
Step 1: Multiply by 13
13 · (3x) = 1
3 · (6)Step 2: Simplify the Right
(13 · 3
)x = 2
Step 3: Simplify the Left 1 · x = 2Step 4: Solution x = 2
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving a Simple Equation
The most basic algebra equation, ax = b, is solved using themultiplicative inverse of a.
Example
Solve the equation 3x = 6 for x .
Step 1: Multiply by 13
13 · (3x) = 1
3 · (6)Step 2: Simplify the Right
(13 · 3
)x = 2
Step 3: Simplify the Left 1 · x = 2Step 4: Solution x = 2
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving a Simple Equation
The most basic algebra equation, ax = b, is solved using themultiplicative inverse of a.
Example
Solve the equation 3x = 6 for x .
Step 1: Multiply by 13
13 · (3x) = 1
3 · (6)Step 2: Simplify the Right
(13 · 3
)x = 2
Step 3: Simplify the Left 1 · x = 2Step 4: Solution x = 2
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving a Simple Equation
The most basic algebra equation, ax = b, is solved using themultiplicative inverse of a.
Example
Solve the equation 3x = 6 for x .
Step 1: Multiply by 13
13 · (3x) = 1
3 · (6)Step 2: Simplify the Right
(13 · 3
)x = 2
Step 3: Simplify the Left 1 · x = 2Step 4: Solution x = 2
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving a Simple Equation
The most basic algebra equation, ax = b, is solved using themultiplicative inverse of a.
Example
Solve the equation 3x = 6 for x .
Step 1: Multiply by 13
13 · (3x) = 1
3 · (6)Step 2: Simplify the Right
(13 · 3
)x = 2
Step 3: Simplify the Left 1 · x = 2Step 4: Solution x = 2
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving a Simple Equation
The most basic algebra equation, ax = b, is solved using themultiplicative inverse of a.
Example
Solve the equation 3x = 6 for x .
Step 1: Multiply by 13
13 · (3x) = 1
3 · (6)Step 2: Simplify the Right
(13 · 3
)x = 2
Step 3: Simplify the Left 1 · x = 2Step 4: Solution x = 2
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Solving a Simple Equation
The most basic algebra equation, ax = b, is solved using themultiplicative inverse of a.
Example
Solve the equation 3x = 6 for x .
Step 1: Multiply by 13
13 · (3x) = 1
3 · (6)Step 2: Simplify the Right
(13 · 3
)x = 2
Step 3: Simplify the Left 1 · x = 2Step 4: Solution x = 2
This solution process worked because 13 is the inverse of 3, so that
13 · 3 = 1, the identity for multiplication.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Outline
1 Solving a Matrix Equation
2 The Inverse of a Matrix
3 Solving Systems of Equations
4 Conclusion
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Matrix Inverse
To solve the matrix equation AX = B we need to find a matrixwhich we can multiply by A to get the identity In.
Matrix Inverse
Let A be an n× n matrix. Then a matrix A−1 is the inverse of A ifAA−1 = A−1A = In.
Caution:
Just as with numbers, not every matrix will have an inverse!
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Matrix Inverse
To solve the matrix equation AX = B we need to find a matrixwhich we can multiply by A to get the identity In.
Matrix Inverse
Let A be an n× n matrix. Then a matrix A−1 is the inverse of A ifAA−1 = A−1A = In.
Caution:
Just as with numbers, not every matrix will have an inverse!
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Matrix Inverse
To solve the matrix equation AX = B we need to find a matrixwhich we can multiply by A to get the identity In.
Matrix Inverse
Let A be an n× n matrix. Then a matrix A−1 is the inverse of A ifAA−1 = A−1A = In.
Caution:
Just as with numbers, not every matrix will have an inverse!
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Verifying Matrix Inverses
Example
Show that
[−1 −23 4
]and
[2 1−3
2 −12
]are inverses.
1
[−1 −23 4
] [2 1−3
2 −12
]=
[1 00 1
]= I2
2
[2 1−3
2 −12
] [−1 −23 4
]=
[1 00 1
]= I2
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Verifying Matrix Inverses
Example
Show that
[−1 −23 4
]and
[2 1−3
2 −12
]are inverses.
1
[−1 −23 4
] [2 1−3
2 −12
]=
[1 00 1
]= I2
2
[2 1−3
2 −12
] [−1 −23 4
]=
[1 00 1
]= I2
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Verifying Matrix Inverses
Example
Show that
[−1 −23 4
]and
[2 1−3
2 −12
]are inverses.
1
[−1 −23 4
] [2 1−3
2 −12
]=
[1 00 1
]= I2
2
[2 1−3
2 −12
] [−1 −23 4
]=
[1 00 1
]= I2
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Verifying Matrix Inverses
Example
Show that
[−1 −23 4
]and
[2 1−3
2 −12
]are inverses.
1
[−1 −23 4
] [2 1−3
2 −12
]=
[1 00 1
]= I2
2
[2 1−3
2 −12
] [−1 −23 4
]=
[1 00 1
]= I2
While it is relatively easy to verify that matrices are inverses, wereally need to be able to find the inverse of a given matrix.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding a Matrix Inverse
To find the inverse of a matrix A we will use the fact thatAA−1 = In.
Find A−1
Let A =
[3 2−1 4
]and find A−1 =
[x1 x2
x3 x4
].
AA−1 = I2 ⇒[
3 2−1 4
] [x1 x2
x3 x4
]=
[1 00 1
][
3x1 + 2x3 3x2 + 2x4
−x1 + 4x3 −x2 + 4x4
]=
[1 00 1
]This gives two systems of equations:{
3x1 + 2x3 = 1−x1 + 4x3 = 0
{3x2 + 2x4 = 0−x2 + 4x4 = 1
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding a Matrix Inverse
To find the inverse of a matrix A we will use the fact thatAA−1 = In.
Find A−1
Let A =
[3 2−1 4
]and find A−1 =
[x1 x2
x3 x4
].
AA−1 = I2 ⇒[
3 2−1 4
] [x1 x2
x3 x4
]=
[1 00 1
][
3x1 + 2x3 3x2 + 2x4
−x1 + 4x3 −x2 + 4x4
]=
[1 00 1
]This gives two systems of equations:{
3x1 + 2x3 = 1−x1 + 4x3 = 0
{3x2 + 2x4 = 0−x2 + 4x4 = 1
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding a Matrix Inverse
To find the inverse of a matrix A we will use the fact thatAA−1 = In.
Find A−1
Let A =
[3 2−1 4
]and find A−1 =
[x1 x2
x3 x4
].
AA−1 = I2 ⇒[
3 2−1 4
] [x1 x2
x3 x4
]=
[1 00 1
][
3x1 + 2x3 3x2 + 2x4
−x1 + 4x3 −x2 + 4x4
]=
[1 00 1
]This gives two systems of equations:{
3x1 + 2x3 = 1−x1 + 4x3 = 0
{3x2 + 2x4 = 0−x2 + 4x4 = 1
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding a Matrix Inverse
To find the inverse of a matrix A we will use the fact thatAA−1 = In.
Find A−1
Let A =
[3 2−1 4
]and find A−1 =
[x1 x2
x3 x4
].
AA−1 = I2 ⇒[
3 2−1 4
] [x1 x2
x3 x4
]=
[1 00 1
][
3x1 + 2x3 3x2 + 2x4
−x1 + 4x3 −x2 + 4x4
]=
[1 00 1
]This gives two systems of equations:{
3x1 + 2x3 = 1−x1 + 4x3 = 0
{3x2 + 2x4 = 0−x2 + 4x4 = 1
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding a Matrix Inverse
To find the inverse of a matrix A we will use the fact thatAA−1 = In.
Find A−1
Let A =
[3 2−1 4
]and find A−1 =
[x1 x2
x3 x4
].
AA−1 = I2 ⇒[
3 2−1 4
] [x1 x2
x3 x4
]=
[1 00 1
][
3x1 + 2x3 3x2 + 2x4
−x1 + 4x3 −x2 + 4x4
]=
[1 00 1
]This gives two systems of equations:{
3x1 + 2x3 = 1−x1 + 4x3 = 0
{3x2 + 2x4 = 0−x2 + 4x4 = 1
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding the Matrix Inverse, Cont. . .
Finding a Matrix Inverse, Continued
Solve the systems of equations:{3x1 + 2x3 = 1−x1 + 4x3 = 0
{3x2 + 2x4 = 0−x2 + 4x4 = 1[
3 2 1 0−1 4 0 1
][1 0 4
14 − 214
0 1 114
314
]
A−1 =
[414 − 2
14114
314
]
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding the Matrix Inverse, Cont. . .
Finding a Matrix Inverse, Continued
Solve the systems of equations:{3x1 + 2x3 = 1−x1 + 4x3 = 0
{3x2 + 2x4 = 0−x2 + 4x4 = 1[
3 2 1 0−1 4 0 1
][1 0 4
14 − 214
0 1 114
314
]
A−1 =
[414 − 2
14114
314
]
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding the Matrix Inverse, Cont. . .
Finding a Matrix Inverse, Continued
Solve the systems of equations:{3x1 + 2x3 = 1−x1 + 4x3 = 0
{3x2 + 2x4 = 0−x2 + 4x4 = 1[
3 2 1 0−1 4 0 1
][1 0 4
14 − 214
0 1 114
314
]
A−1 =
[414 − 2
14114
314
]
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding the Matrix Inverse, Cont. . .
Finding a Matrix Inverse, Continued
Solve the systems of equations:{3x1 + 2x3 = 1−x1 + 4x3 = 0
{3x2 + 2x4 = 0−x2 + 4x4 = 1[
3 2 1 0−1 4 0 1
][1 0 4
14 − 214
0 1 114
314
]
A−1 =
[414 − 2
14114
314
]
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
General Rule for Finding An Inverse
Applying the lessons of the previous example yields a generalprocedure for finding the inverse of a matrix.
Finding a Matrix Inverse
To find the inverse of a n× n matrix A, form the augmented matrix[A | In] and use row reduction to transform it into
[In | A−1
].
Caution:
It may not be possible to get In on the left side of the matrix. If itis not possible, then the matrix A has no inverse.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
General Rule for Finding An Inverse
Applying the lessons of the previous example yields a generalprocedure for finding the inverse of a matrix.
Finding a Matrix Inverse
To find the inverse of a n× n matrix A, form the augmented matrix[A | In] and use row reduction to transform it into
[In | A−1
].
Caution:
It may not be possible to get In on the left side of the matrix. If itis not possible, then the matrix A has no inverse.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
General Rule for Finding An Inverse
Applying the lessons of the previous example yields a generalprocedure for finding the inverse of a matrix.
Finding a Matrix Inverse
To find the inverse of a n× n matrix A, form the augmented matrix[A | In] and use row reduction to transform it into
[In | A−1
].
Caution:
It may not be possible to get In on the left side of the matrix. If itis not possible, then the matrix A has no inverse.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding the Inverse of a 3× 3 Matrix
Example
Find the inverse of the following matrix, if it exists.1 1 13 −4 20 0 0
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Finding the Inverse of a 3× 3 Matrix
Example
Find the inverse of the following matrix, if it exists.1 1 13 −4 20 0 0
Row operations yield:1 0 6
747
17 0
0 1 17
37 −1
7 00 0 0 0 0 1
And therefore there is no inverse.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Outline
1 Solving a Matrix Equation
2 The Inverse of a Matrix
3 Solving Systems of Equations
4 Conclusion
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Using A−1 to Solve a System of Equations
The main reason we are interested in matrix inverses is to solve asystem of equations written in matrix form.
Solving a System of Equations
If A is the matrix of coefficients for a system of equations, X is thecolumn vector containing the system variables, and B is thecolumn vector containing the constants, then:
AX = B ⇒ A−1AX = A−1B
⇒ InX = A−1B
⇒ X = A−1B
Caution:
A system of equations can only be solved in this way if it has aunique solution.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Using A−1 to Solve a System of Equations
The main reason we are interested in matrix inverses is to solve asystem of equations written in matrix form.
Solving a System of Equations
If A is the matrix of coefficients for a system of equations, X is thecolumn vector containing the system variables, and B is thecolumn vector containing the constants, then:
AX = B ⇒ A−1AX = A−1B
⇒ InX = A−1B
⇒ X = A−1B
Caution:
A system of equations can only be solved in this way if it has aunique solution.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Using A−1 to Solve a System of Equations
The main reason we are interested in matrix inverses is to solve asystem of equations written in matrix form.
Solving a System of Equations
If A is the matrix of coefficients for a system of equations, X is thecolumn vector containing the system variables, and B is thecolumn vector containing the constants, then:
AX = B ⇒ A−1AX = A−1B
⇒ InX = A−1B
⇒ X = A−1B
Caution:
A system of equations can only be solved in this way if it has aunique solution.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
An Example
Example
Use a matrix equation to set-up and solve each system ofequations given below.
1
{−x − 2y = 13x + 4y = 3
2
x + y − z = 63x − y = 82x − 3y + 4z = −3
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
An Example
Example
Use a matrix equation to set-up and solve each system ofequations given below.
1
{−x − 2y = 13x + 4y = 3
2
x + y − z = 63x − y = 82x − 3y + 4z = −3
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
An Example
Example
Use a matrix equation to set-up and solve each system ofequations given below.
1
{−x − 2y = 13x + 4y = 3
2
x + y − z = 63x − y = 82x − 3y + 4z = −3
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Reusing Results
One major advantage of solving a system of equations using amatrix equation is that if your matrix of coefficients A stays thesame, but your matrix B changes, you can reuse most of your work.
Example
Solve each of the following systems of equations using the resultsfrom the last part of the previous question.
1
x + y − z = 23x − y = 12x − 3y + 4z = 0
2
x + y − z = 03x − y = −142x − 3y + 4z = −13
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Reusing Results
One major advantage of solving a system of equations using amatrix equation is that if your matrix of coefficients A stays thesame, but your matrix B changes, you can reuse most of your work.
Example
Solve each of the following systems of equations using the resultsfrom the last part of the previous question.
1
x + y − z = 23x − y = 12x − 3y + 4z = 0
2
x + y − z = 03x − y = −142x − 3y + 4z = −13
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Reusing Results
One major advantage of solving a system of equations using amatrix equation is that if your matrix of coefficients A stays thesame, but your matrix B changes, you can reuse most of your work.
Example
Solve each of the following systems of equations using the resultsfrom the last part of the previous question.
1
x + y − z = 23x − y = 12x − 3y + 4z = 0
2
x + y − z = 03x − y = −142x − 3y + 4z = −13
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Reusing Results
One major advantage of solving a system of equations using amatrix equation is that if your matrix of coefficients A stays thesame, but your matrix B changes, you can reuse most of your work.
Example
Solve each of the following systems of equations using the resultsfrom the last part of the previous question.
1
x + y − z = 23x − y = 12x − 3y + 4z = 0
2
x + y − z = 03x − y = −142x − 3y + 4z = −13
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Outline
1 Solving a Matrix Equation
2 The Inverse of a Matrix
3 Solving Systems of Equations
4 Conclusion
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Important Concepts
Things to Remember from Section 2-6
1 A−1A = AA−1 = In for an n × n matrix A
2 Find A−1 by reducing [A | In] to [In | A]
3 Solving systems of equations with matrix equations.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Important Concepts
Things to Remember from Section 2-6
1 A−1A = AA−1 = In for an n × n matrix A
2 Find A−1 by reducing [A | In] to [In | A]
3 Solving systems of equations with matrix equations.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Important Concepts
Things to Remember from Section 2-6
1 A−1A = AA−1 = In for an n × n matrix A
2 Find A−1 by reducing [A | In] to [In | A]
3 Solving systems of equations with matrix equations.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Important Concepts
Things to Remember from Section 2-6
1 A−1A = AA−1 = In for an n × n matrix A
2 Find A−1 by reducing [A | In] to [In | A]
3 Solving systems of equations with matrix equations.
Solving a Matrix Equation The Inverse of a Matrix Solving Systems of Equations Conclusion
Next Time. . .
Next time we will review for our third and final in class exam. Itwill be over the sections we covered from chapters 1 and 2.
For next time
Review for Exam