# Computational efficiency using Gaussian elimination

Assume it took 2 seconds to solve $$Ax=b$$ for $$x$$ (where $$A$$ is a $$3 \times 3$$ matrix and $$b$$ is a $$3 \times 1$$ matrix) using Gaussian elimination, how much longer would it take to:

a) use Gaussian elimination to find $$A^{−1}$$ and then find $$x = A^{−1}\cdot b$$

b) if $$A$$ were a $$30 \times 30$$ matrix and $$b$$ were a $$30 \times 1$$ matrix and I used Gaussian elimination to find $$x$$

c) if $$A$$ were a $$30 \times 30$$ matrix and $$b$$ were a $$30 \times 1$$ matrix and I used Gaussian elimination to find $$A^{−1}$$ and then find $$x$$ from $$x=A^{−1} \cdot b$$

To get started, I know that I will need to use $$\frac 23n^3$$ where n is the number of operations. I know this article touches on it. But what exactly do I need to do? How many operations does it take to find $$A^{-1}$$ and then the operation $$A^{-1}\cdot b$$? How could i figure this out?

The efficiency of Gauss elimination is $\mathcal O (n^3)$, so it should take 2000 seconds to do a 30x30 matrix.