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Matrix inverse using linear algebra

Consider the set of equations:

3x + 4y = 7
6x + 4y = 10

These can be represented by a matrix as:

AX = C

This can be solved by using the following operations:

Basically, take a row and eliminate one of its columns by subtracting a suitable multiple of any one row.
Then move onto other row and eliminate some other cell.

For a matrix with NxN cells and N values, N cells will remain after all such transformations.
Then each value can be found easily by normalizing each number-pair in each row as shown above.

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