Lesson 23 - 4 05-Inverse Matrix PDF

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Summary

Computer Math Fundamentals...

Description

Identity Matrix The identity matrix, In , is a special matrix with a value of 1 along the diagonal elements (akk where 0 < k ≤ n) and a value of 0 for all other elements. Example:

 =  

   

Multiplying any m x n matrix A by the identity matrix I n has a product of A.

= Example:

 = 

  

    

 = 

  =   

 = 

Inverse Matrix Given an n x n matrix A, the inverse A -1 is an n x n matrix such that AA-1 = A-1A = In Example:

Given

 = 

 = 

  

  − 

 = −

−  ( =   (

−   , is B the inverse of A? 

)( ) + ( )(− ) ( )(− ) + ( )( ) =  )( ) + ( )( − ) ( )( − ) + ( )( )  

  

Since AB = I2 , B is the inverse of A

MATH18584

Lesson 05: Inverse Matrix

1

Finding the Inverse of a 2 x 2 Matrix

 = 

For any 2 x 2 matrix:

  , the inverse is: 

  =  

−

− −

− − −

−

  .  

Proof:

−

 = 

− −

  −  −   −

−

   (  =     (   =     =  

−

  =   

The value of

MATH18584

 = 

− −

  

( ) 

− −

 +( 

) 

) 

−

 + ( 

) 

− −

  

( )

− −

 + ( 

)



+ − − + −

    

− − − −

 

−

− −





 

− −

    

 = 

 -1  , find A :  − −

− − −

−

−

) 



 =  Example: If

−

 +( 

) 

=

−

    =   −   

−    =   − 

−   

= is called the determinant of the matrix A.

Lesson 05: Inverse Matrix

2

     

Determinant

In general, for any 2 x 2 matix

 = 

  , the determinant of A is: 

The inverse A-1 can be computed using the determinant of A:

Example: If

MATH18584

 = 

−

=(

=

 ⋅ −

)− ( ) −   

 -1  , compute A using determinants: 

Lesson 05: Inverse Matrix

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Exercises: Compute the determinant and the inverse of the following matrices:

 = 

  

 = 

  

− = 

  

MATH18584

Lesson 05: Inverse Matrix

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Answers: Compute the determinant and the inverse of the following matrices:

 = 

  

det(A) = 4-6 = -2

A-1 =

 ⋅ − −

−  − =   

−

 = 

  

det(B) = 6-7 = -1

B-1 =

 ⋅ − −

−  −  =  

 − 

det(C) = -2-6 = -8

C-1 =

 ⋅ − −

−  − = −  

− = 

  

MATH18584

Lesson 05: Inverse Matrix

  

  

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