To solve the system of equations:
$2x-3y-4z = 2$
$-z = 5$
$x-2y+z = 3$
you could use matrices, the two primary methods I have come across involve using the inverse (method 1) or just using Gaussian elimination straight away on an augmented matrix (method 2). I have attached the two methods below:
My question is whether there is any advantage of using method 1, as you have to carry out the same process as method 2 of Gaussian elimination, but then also have to multiply matrices afterwards, meaning method 2 is surely always quicker? Thanks.
Best Answer
If all you want is to solve $A \vec{x} = \vec{b}$, where $A$ is an invertible matrix, by either
1) Determining $A^{-1}$ using Gaussian elimination on $A$ augmented with the identity matrix, then computing $A^{-1} \vec{b}$, or
2) Using Gaussian elimination on $A$ augmented with $\vec{b}$,
then yes. Method two is the more straightforward method. However, I can still think of several reasons for teaching and learning method one.
Probably the main benefit of method one is that you have $A^{-1}$ to work with at the end of your problem. Suppose your teacher gives you several problems with the same matrix but different $\vec{b}$ vectors. Using method one you only have to use Gaussian elimination once, to find $A^{-1}$. After that all you have to do is compute $A^{-1} \vec{b}$ for each different $\vec{b}$.