I am currently studying linear algebra at university.
We are currently doing Inverses of matrices using Gauss-Jordan Reduction, but no one has really explained how you are supposed to decide what ERR operations to do next.
For example, we have been asked to find the inverse of this 4x4 augmented matrix (matrix on left, identity matrix on right):
But no matter what i do, i cant seem to get the left looking like the identity matrix. I have spoken to others in the class, and they dont know either.
If someone could please explain how you are supposed to decide on the ERR operations to perform next, i would be very grateful.
Thanks Corey
The trick is to focus on one column at a time, starting from the left. So the first thing I'd do is take the 2nd row and multiply it by $1/3$. Adding that row to all the others makes every other entry in the first column $0$. Then you can finish up by multiplying that row by $-1$ and row swapping to make it the first row. The first column now matches the first column of the identity matrix, as desired.
From there, start working on the second column. Choose a row other than the first row and use it to "zero" all the other entries in the second column like we did above. After that, multiply the chosen row by an appropriate constant to turn its leading entry into $1$. Finally, row swap to make that row the second row, and notice that your second column now matches the second column of the identity matrix.
Continue in this manner until you've dealt with all four columns.