help with inverse of the matrix

Question

I need help with finding the inverse of the matrix for,

 

      -9   5

 

       6  -3

 

 

I started with 2 x 2 then    A =  -3/(-9)(-3) - (5)(6)              5/(-9)(6) - (5)(6)

 

 

                                                -6/(-9)(-3) - (5)(6)              -9/(-9)(-3) - (5)(6)

 

 

                                               -3/27-11                          5/-54 - 30

 

                                              -6/27 - 30                          -9/27 - 30

 

 

 

?????

 

 

thanks

Answer 1

There are many ways to find the inverse of a matrix. Without knowing what you know/don't know, it is hard to say how you're going to do it. I will outline a few ways then describe the way I typically show a student for the 2 x 2 case. (assume the inverse exists)

 

1. Augment the matrix with the identity matrix then but the matrix on the left in reduced row echlon form (also known as Gauss Jordan Form) the inverse of the matrix appears on the right.

 

2. Use the determinant and cofactors of the matrix. Then one constructs the adjoint matrix.

 

3. Here is the way I show students:

 

For a 2 x2 matrix ( ( a , b ) , ( c , d ) ) (in this notation, 'a' and 'b' are the first and second entries in the first row and 'c' and 'd' are the entries in the second row)

 

The inverse is

( ( d/(ad-bc) , -b/(ad-bc) ) , ( -c/(ad-bc)  , a/(ad-bc) ) )

 

Perhaps written a bit 'nicer' as

 

1/(ad-bc)  ( ( d , -b ) , ( -c , a ) )

 

So the inverse of the matrix here is

 

( ( 1 , 5/3 ) , ( 2 , 3 ) )