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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?

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

Inverse of a matrix should take double time. So 4 sec for a) and 4000 sec for c).

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