being in matrices Let A B X Y and Z be square matrices of th

being in matrices. Let A, B, X, Y and Z be square matrices of the same size, with A Consider the following partitioned matrices: A B (a) Given that N is MN: I T Ax o A X p A AY BA (b) Using part (a), compute the inverse of M in the case where 1 0 0 1 1 0 -2 Page 6 of 10

Solution

The inverse of matrix M is matrix N as given in part (a).So to find the matrix N we have to find X,Y and Z.As we know from part (a) X is equal to the inverse of matrix A.So let\'s find the inverse of A first.

Determinant of A = 1((1)(2)-(-1)(1))-0((-1)(2)-(0)(1))-2((-1)(-1)-(0)(1)) = 3-0-2 = 1

Determinant is not zero, therefore inverse matrix exists.

Write the augmented matrix : 1 0 -2 1 0 0

                                           -1 1 1 0 1 0

                                           0 -1 2 0 0 1

Find the pivot in the 1st column in the 1st row,which is 1

Multiply the 1st row by -1 : -1 0 2 -1 0 0

                                        -1 1 1 0 1 0

                                         0 -1 2 0 0 1

Subtract the 1st row from the 2nd row and restore it : 1 0 -2 1 0 0

                                                                             0 1 -1 1 1 0

                                                                             0 -1 2 0 0 1

Find the pivot in the 2nd column in the 2nd row,which is 1

Multiply the 2nd row by -1 : 1 0 -2 1 0 0

                                        0 -1 1 -1 -1 0

                                        0 -1 2 0 0 1

Subtract the 2nd row from the 3rd row and restore it : 1 0 -2 1 0 0

                                                                             0 1 -1 1 1 0

                                                                             0 0 1 1 1 1

Find the pivot in the 3rd column in the 3rd row,which is 1

Multiply the 3rd row by -2 : 1 0 -2 1 0 0

                                       0 1 -1 1 1 0

                                       0 0 -2 -2 -2 -2

Subtract the 3rd row from the 1st row and restore it : 1 0 0 3 2 2

                                                                            0 1 -1 1 1 0

                                                                            0 0 1 1 1 1

Multiply the 3rd row by -1 : 1 0 0 3 2 2

                                        0 1 -1 1 1 0

                                       0 0 -1 -1 -1 -1

Subtract the 3rd row from the 2nd row and restore it : 1 0 0 3 2 2

                                                                             0 1 0 2 2 1

                                                                             0 0 1 1 1 1

There is the inverse matrix on the right : 3 2 2

                                                          2 2 1

                                                          1 1 1

Now to find Y we have to find -B,that is multiply each element of B by -1

-1 0 0

0 1 0

0 0 -2

Now Z is just A

So the inverse matrix N is : X Y

                                        0 Z