K 0 1 1 1 1 1 1 1 0 is known to be nonsingular Compute K1 a

K = (0 1 -1 1 -1 1 -1 1 0) is known to be non-singular. Compute K^-1 as a product of elementary matrices. You may use the Inversion algorithm for this purpose.

Solution

First, I write down the entries the matrix K, but I write them in a double-wide matrix:

In the other half of the double-wide, I write the identity matrix:

0 1 -1 ,1 0 0

1 -1 1 , 0 1    0

-1   1   0 , 0     0   1

Now I\'ll do matrix row operations to convert the left-hand side of the double-wide into the identity.

R1-->R1+R2 , R3----> R3+R2

1    0 0 , 1   1    0

1   -1 1 , 0    1    0

0     0     1 , 0   1    1

R2-----> -(R2-(R1+R3))

1   0   0 , 1 1 0

0   1   0 , 1   1   1

0 0    1 , 0   1    1

Now that the left-hand side of the double-wide contains the identity, the right-hand side contains the inverse. That is, the inverse matrix is the following:

1 1 0

1 1 1

0 1 1