In this video I run through the thinking behind Gaussian Elimination by working through an example. Contrary to my belief when I was first introduced to it at University, you don't have to have experience in solving problems with this to get to an answer, just follow the algorithm and you'll get to your solution.
A few points I didn't mention in the video:
1) The elements that we use to cancel out elements (Row,Col) -- (2,1) , (3,1) , (3,2) are the *pivot elements of the matrix. For an element to be a pivot element: it has to be the first non-zero element in each row and all elements below that element are zero; it also has to be to the right of the pivot element in the row above.
*usually pivot elements will have been normalised to 1 but in my opinion the 'pivot' name is more to do with its' location in the matrix than its' value. It's a good way to explain to someone where to look in a matrix.
2) We could have continued the Gaussian Elimination method to eliminate the upper triangle of the matrix and leave ourselves with just the leading diagonal of elements, effectively solving it whilst in matrix algebra form, instead of substituting the variables back through the equations. After normalising the leading diagonal elements, the result is called the 'reduced row echelon' form. I'll do a video on this some other time.
3) If any of the leading diagonal elements end up equalling zero then there is not a unique solution. It's important to organise the matrix so that the elements along the diagonal are non-zero because it has the potential to mess up the algorithm if they are.
Intro music by an amazing band called Wordy, check them out here:
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