I would like to check if what I have done is correct. Please, any input is appreciated.
**Problem statement:** Consider a non-singular matrix . Construct an algorithm using Gaussian elimination to find Provide the operation counts for this algorithm.
**My attempted algorithm and operations count:**
Let be an augmented $n$ x $2n$ matrix.
INPUT: number of unknowns and equations n, augmented matrix
OUTPUT: , provided that the inverse exists
STEP 1: For do STEPS 2-4
STEP 2: Let $p$ be the smallest integer with such that . If no integer $p$ exists then output ('no unique solution exists'); STOP
STEP 3: If then perform
STEP 4: For do STEPS 5-6
STEP 5: Set .
STEP 6: Perform
STEP 7: If then output NO UNIQUE SOLUTION EXISTS and STOP
STEP 8: For ,
For do STEPS 9 and 10
STEP 9: Set
STEP 10: Perform
STEP 11: for
STEP 12: OUTPUT $A^{-1}$ and STOP.
**I am getting the operations count as follows:** A total of $(2n^3+9n^2+n)/3$ multiplications and divisions and a total of $(2n^3+6n^2-8n)/3$ additions and subtractions for a grand total of $4n^3/3 + 5n^2 - 7n/3$ operations. **Does this sound about right?**
Thanks for any help.