How can ratios in gauss-jordan elimination?
Okay, I found one with a lot of ratios. The objective is to invert the matrix using a specific theorem in the textbook and as you will see, there is a lot of ratios, nearly everything becomes a ratio. Here I go:
Q4. Matrix A=
5 3 2 : 4
3 3 2 : 2
0 1 1 : 5
The theorem basically uses an identity matrix and applies the same operations to reduce the original matrix and the identity matrix then becomes the inverted matrix, so the format below is
Matrix A : Identity matrix : B
5 3 2 : 1 0 0 : 4 (row I * 1/5
3 3 2 : 0 1 0 : 2
0 1 1 : 0 0 1 : 5
1 3/5 2/5 : 1/5 0 0 : 4/5
3 3 2 : 0 1 0 : 2
0 1 1 : 0 0 1 : 5
I went further and reduced the first column and realize 95% of it has become ratios, there must be a simpler way to work this through. Please help.
Well, if the inverse matrix has non-integer entries, there is no way you can avoid getting them at some point during the process.
In fact, I think that if you start with a matrix having integer entries, the only way you can invert it using the Gauss-Jordan algorithm without ever seeing non-integers in the process is if the determinant of the matrix is $\displaystyle \pm 1$.