Difficulty with Gaussian Elimination involving a,b coefficients

Question

I have the system matrix as follows:

{1 2 2 1

1 a 3 3

1 11 a b}

I am attempting to row reduce this matrix, but am having difficulty in this process, particularly with Row 3. I come to

{1 2 2 1

0 a-2 1 2

0 9 a-2 b-1}

but now I cannot see how to resolve the 9 in the third row to a zero.

My goal in all this is to determine what a must be (or must not be) in order for the system to have a unique solution.

Thanks for your help!

Answer 1

It has been a while since I last opened my LA book, but I think it's something like this, right?

Replace a - 2 by x and b by y because it simplifies the whole thing.

0 9 x y

Now divide the third row by 9.

0 1 (x / 9) (y / 9)

Then multiply it with x (or a - 2).

0 x (x ^ 2 / 9) (x * y / 9)

Now you can subtract the second row.

0 0 (x ^ 2 / 9 - 1) (x * y / 9 - 2)