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A system of equations may be written as a matrix equation. <= /div>
An Application of Inverse Matrices
------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
A system of equations may be written as a matrix equation. To solve the sy= stem
An Application of Inverse Matrices
ax + by =3D k
cx + dy =3D m
is the same as solving fo= r x and y in the equation
ax + by  
cx + dy
=3D
k
m
.
------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAzAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB ADIAgAAAAAAAAAIGjI+py50FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAK AAEAgAAAAAAAAAIDjI8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAK AAEAgAAAAAAAAAIDjI8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAzAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB ADIAgAAAAAAAAAIGjI+py50FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAK AAEAgAAAAAAAAAIDjI8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAzAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB ADEAgAAAAAAAAAIGjI+py40FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAK AAEAgAAAAAAAAAIDjI8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAzAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB ADEAgAAAAAAAAAIGjI+py40FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0474_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0475.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
A system of equations may be written as a matrix equation. To solve the sy= stem
An Application of Inverse Matrices
ax + by =3D k
cx + dy =3D m
is the same as solving fo= r x and y in the equation
ax + by  
cx + dy
=3D
k
m
.
If we set
a    b  
c    d
as A, 
x
y
as X, and
k
m
as B,
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file:///C:/2F561913/6InverseMatrices_files/slide0475_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAzAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAB ADIAgAAAAAAAAAIGjI+py50FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0475_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAK AAEAgAAAAAAAAAIDjI8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0475_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAK AAEAgAAAAAAAAAIDjI8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0476.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
A system of equations may be written as a matrix equation. To solve the sy= stem
An Application of Inverse Matrices
ax + by =3D k
cx + dy =3D m
is the same as solving fo= r x and y in the equation
ax + by  
cx + dy
=3D
k
m
.
then
ax + by  
cx + dy
=3D
a    b  
c    d
x
y
If we set
a    b  
c    d
as A, 
x
y
as X, and
k
m
as B,
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file:///C:/2F561913/6InverseMatrices_files/slide0477.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
A system of equations may be written as a matrix equation. To solve the sy= stem
An Application of Inverse Matrices
ax + by =3D k
cx + dy =3D m
is the same as solving fo= r x and y in the equation
ax + by  
cx + dy
=3D
k
m
.
then
ax + by  
cx + dy
=3D
a    b  
c    d
x
y
If we set
a    b  
c    d
as A, 
x
y
as X, and
=3D AX,
k
m
as B,
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file:///C:/2F561913/6InverseMatrices_files/slide0478.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
A system of equations may be written as a matrix equation. To solve the sy= stem
An Application of Inverse Matrices
ax + by =3D k
cx + dy =3D m
is the same as solving fo= r x and y in the equation
ax + by  
cx + dy
=3D
k
m
.
then
ax + by  
cx + dy
=3D
a    b  
c    d
x
y
If we set
a    b  
c    d
as A, 
x
y
as X, and
=3D AX,
and the system is the same as AX =3D B.
k
m
as B,
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file:///C:/2F561913/6InverseMatrices_files/slide0479.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
A system of equations may be written as a matrix equation. To solve the sy= stem
An Application of Inverse Matrices
ax + by =3D k
cx + dy =3D m
is the same as solving fo= r x and y in the equation
ax + by  
cx + dy
=3D
k
m
.
then
ax + by  
cx + dy
=3D
a    b  
c    d
x
y
If we set
a    b  
c    d
as A, 
x
y
as X, and
=3D AX,
and the s= ystem is the same as AX =3D B. Solving for x
and y is = the same as solving for the matrix X  = .
k
m
as B,
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file:///C:/2F561913/6InverseMatrices_files/slide0482.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Example B: <= /span>
a. Write the following system as a matrix equation
x + 2y =3D = 4
x + 3y =3D = -1
An Application of Inverse Matrices
------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0419.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Example B: <= /span>
a. Write the following system as a matrix equation
x + 2y =3D = 4
x + 3y =3D = -1
The system = may be expressed as
1     2
1     3
x
y
=3D<= /div>
4
-1
An Application of Inverse Matrices
------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0480.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Example B: <= /span>
a. Write the following system as a matrix equation
If we let A =3D 
1     2
1     3
x + 2y =3D = 4
x + 3y =3D = -1
, X =3D
x
y
,
The system = may be expressed as
1     2
1     3
x
y
=3D<= /div>
4
-1
then the ma= trix equation is simply AX =3D B.
=
and B =3D
4
-1
.
An Application of Inverse Matrices
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file:///C:/2F561913/6InverseMatrices_files/slide0480_image258.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAzAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB ADEAgAAAAAAAAAIGjI+py40FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0480_image259.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0481.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Example B: <= /span>
a. Write the following system as a matrix equation
If we let A =3D 
1     2
1     3
x + 2y =3D = 4
x + 3y =3D = -1
, X =3D
x
y
,
The system = may be expressed as
1     2
1     3
x
y
=3D<= /div>
4
-1
then the ma= trix equation is simply AX =3D B.
=
and B =3D
4
-1
.
An Application of Inverse Matrices
The solve the matrix X, we use the following inverse <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>matrix method.
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file:///C:/2F561913/6InverseMatrices_files/slide0481_image283.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAzAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB ADEAgAAAAAAAAAIGjI+py40FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0481_image284.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0420.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
To solve the  equation AX =3D B for the matrix = X,
we multiply to the left of both sides by A-1 (if it exists) 
An Application of Inverse Matrices
------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0483.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
To solve the  equation AX =3D B for the matrix = X,
we multiply to the left of both sides by A-1 (if it exists) 
AX =3D B
An Application of Inverse Matrices
------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0484.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
To solve the  equation AX =3D B for the matrix = X,
we multiply to the left of both sides by A-1 (if it exists) 
AX =3D B
A-1AX =3D A-1B
An Application of Inverse Matrices
------=_NextPart_01C8AF1D.4B102200 Content-Location: file:///C:/2F561913/6InverseMatrices_files/slide0485.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
To solve the  equation AX =3D B for the matrix = X,
we multiply to the left of both sides by A-1 (if it exists) 
AX =3D B
A-1AX =3D A-1B
IX =3D A-1B
or that X =3D A-1B
An Application of Inverse Matrices
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To solve the  equation AX =3D B for the matrix = X,
we multiply to the left of both sides by A-1 (if it exists) 
AX =3D B
A-1AX =3D A-1B
IX =3D A-1B
or that X =3D A-1B
b. Solve the matrix equation AX =3D B from part a. by the inverse matrix method.
An Application of Inverse Matrices
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To solve the  equation AX =3D B for the matrix = X,
we multiply to the left of both sides by A-1 (if it exists) 
AX =3D B
A-1AX =3D A-1B
IX =3D A-1B
or that X =3D A-1B
b. Solve the matrix equation AX =3D B from part a. by the inverse matrix method.
from previo= us example we've that   
Given that A =3D 
1     2
1     3
3    -2
-1     1   .=
A-1 = =3D 
An Application of Inverse Matrices
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