Suppose T is the product of all elementary transformations and applying it to A, this gives us:
The gaussian elimination algorithm uses only the third form of the three elementary matrix transformations to make TA upper triangular. Using the properties of these elementary transformations, we calculate the inverse of T which shows us it is a lower triangular matrix L which can be added to the left and right side of the previous equation resulting in:
Solving Once the upper and lower triangular matrices are known, solving this equation becomes very efficient as it can be split up into:
Applications The matrices L and U can be used to calculate the matrix inverse. Computer implementations that invert matrices often use this approach.
See also
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