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So, what's all this fuss about inverse matrices?

You can use them to solve systems of equations.

Check it out:

2x - 5y = 8

                                      3x + 4y = -6
       Let's get that coefficient matrix:

A = [ row 1: 2 , -5  row 2: 3 , 4 ]

       Then, we'll need matrices for

X = [ row 1: x  row 2: y ]B = [ row 1: 8  row 2: -6 ]
our variablesour "= guys"

In short, this system can be rewritten as

AX = B
 

and we can use algebra to solve it!

AX = B
 

A^( -1 ) times AX = A^( -1 ) times B ... X = A^( -1 ) times B ... order REALLY matters here!