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Exact inversion of matrix complexity (by Gaussian elimination)


Oct 7, 2013
I would like to check if what I have done is correct. Please, any input is appreciated.

**Problem statement:** Consider a non-singular matrix $A_{nxn}$. Construct an algorithm using Gaussian elimination to find $A^{-1}$. Provide the operation counts for this algorithm.

**My attempted algorithm and operations count:**

Let $[A|I]$ be an augmented $n$ x $2n$ matrix.

INPUT: number of unknowns and equations n, augmented matrix

OUTPUT: $A^{-1}$, provided that the inverse exists

STEP 1: For $i=1,\dots,n-1$ do STEPS 2-4

STEP 2: Let $p$ be the smallest integer with $i \leq p \leq n$ such that $a_{pi} \neq 0$. If no integer $p$ exists then output ('no unique solution exists'); STOP

STEP 3: If $p \neq i$ then perform $E_p \leftrightarrow E_i$

STEP 4: For $j=i+1, \dots, n$ do STEPS 5-6

STEP 5: Set $m_{ji}=a_{ji}/a_{ii}$.

STEP 6: Perform $(E_j - m_{ji}E_i) \rightarrow (E_j)$

STEP 7: If $a_{nn}=0$ then output NO UNIQUE SOLUTION EXISTS and STOP

STEP 8: For $i=n-1,n-2,\dots ,1$,
For $j=i+1,i,\dots ,1$ do STEPS 9 and 10

STEP 9: Set $m_{ij}=a_{ij}/a_{jj}$

STEP 10: Perform $(E_i - m_{ij}E_j) \rightarrow (E_i)$

STEP 11: for i=1, ldots, n, do $E_i/a_{ii} \rightarrow E_i$

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.