# Finding Inverse Matrix by Definition

Hi fellow mathies,

So, I'm wondering if the way to find the inverse of a matrix by definition (instead of using a special algorithm/tacking on the identity matrix and reducing, etc), is to multiply the matrix by a variable matrix and have it equal the identity matrix.

So, for example:

this matrix

3 0 0
0 -1 3
0 -3 -1

multiplied by this matrix

a b c
d e f
g h i

would equal

1 0 0
0 1 0
0 0 1

would be

entry a is 3a+0+0=1
entry b is 3b+0+0=0
entry c is 3c + 0 + 0 =0
entry d is 0 -d + 3g = 0
entry 3 is 0 -e + +3h = 1
etc.

and then solve by substitution and basic algebra.

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