Find the inverse of AB if

Question

Find the inverse of AB if 

 

A^-1 = 3 4

       -5 1

 

B^-1 = -3 5 

        4 -1

 

(AB)^-1=

 

Answer 1

There may be an easier way to do this, but here goes:

 

If matrix A = [a   b ]

                     c   d

                    

The inverse of a matrix is given by

A^-1 = 1/detA [ d  -b ]

                     -c   a 

 

Where the detA = (a*d - b*c)

If we go backwards from A^-1 to get A, then A = 23* [ 1   -4 ]

                                                                              5    3

detA = (1*3-(-4*5)) = (3+20) = 23

We can check that we calculated A correctly by determining A^-1 from A:  (1/23) * 23 * [ 3    4]

                                                                                                                               -5    1

Reducing: A^-1 =  [ 3   4]
                           -5  1 

 

Now we do the same with B^-1 to get B = 23 * [-1   -5 ]

                                                                    -4   -3

You perform matrix multiplication to get AB:

(AB) = 23*A*23*B = 529*(AB)

529 *[ 3    4]  *  [-1   -5 ]
          -5    1        -4   -3

529 * [3*-1+4*-4        3*-5+4*-3]

          -5*-1+1*-4       5*-5+1*-3

529* [-19   -27]

          -9    -28

 

Now to get (AB)^-1, you take the inverse of matrix AB:

(AB)^-1 = 529* (1/det(AB)) *[-28     27 ]

                                             9     -19

det(AB) = (-19*-28 - (-27*-9)) = 289

 

(AB)-1 = (529/289) *[-28    27 ]
                                  9     -19