Finding Inverse Matrix by Definition

[shelovesmath]

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.

Thanks in advance :)

 

[Fredrik]

Yes, that works. (Is that all you wanted to know?). I just wouldn't call the entries "a,b,c,d,3,...". I actually don't understand your labeling system (unless you meant e when you wrote 3). You should refer to the entries by row number and column number. For example, what you called "3" is row 2, column 2, i.e. the "22" entry of the product.