martrics question

Question

Under the bracket of matrices ..

 1/2.   2/5

2/5.     -1/5

     

 

Find its inverse..

 

Answer 1

Sonu,

 

Are you familiar with a general formula for finding the inverse of a matrix?  You can take these steps to find the inverse:

 

(1) Replace each matrix element with its minor.  The minor is the determinant of the smaller matrix that remains after you delete the row and column of the matrix element.  This is called the matrix of the minors.

 

(2) Multiply each minor by (-1)^i+j, where i and j are the row and column, respectively, where the minor is located.  Note (-1)^i+j alternates in sign as you move across a row or down a column. The resulting matrix is called the cofactor matrix

 

(3) Take the transpose of the cofactor matrix.  The resulting matrix is called the adjoint matrix

 

(4) Divide the adjoint matrix by the determinant of the original matrix.  The resulting matrix is the inverse of the matrix you started with.

 

The steps in Andrew's solution are equivalent to this.  For a 2x2 matrix, this is easy.  For 3x3 and higher, it gets tedious, unless you have a programmable tool to work with.  There are other ways to find the inverse, but working with the adjoint matrix is very common.

 

 

Answer 2

The inverse of matrix A

 

Matrix A

----------

 | a    b |

 | c    d |

 

Matrix A^-1

-------------

   

     1       | d -b |   

 --------  |-c   a |

  ad-bc

 

Switch the position of a, d

Change sign of b, c

Multiply each term by 1/(ad-bc)

 

| 1/2   2/5 |

| 2/5  -1/5 |

 

 

1/[(1/2)(-1/5)-(2/5)(2/5)] |-1/5   -2/5 |

                                       |-2/5    1/2 |

 

1/(-1/10 - 4/25) |-1/5   -2/5 |

                         |-2/5    1/2 |

 

1/(-13/50) |-1/5   -2/5 | 

                 |-2/5    1/2 |

 

-50/13 |-1/5   -2/5 | 

           |-2/5    1/2 |

 

| -1/5(-50/13)  -2/5(-50/13) |

| -2/5(-50/13)    1/2(-50/13)|

 

| 10/13   20/13 |

| 20/13   -25/13|