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Inverse Matrices

If we have a matrix

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

We can't write 1 / A -- a result would require division.

So...  Can we find  A^( -1 )?

We sure can!  It's called an inverse matrix.  Here's how you find it:

Let's start with this matrix

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

 

This is going to work a lot like Gaussian elimination.  (If you've ever seen that before.)

We make a big double matrix

[ row 1: 2 , 3 row 2: -4 , -5 | row 1: 1 , 0 row 2: 0 , 1 ]
A on this side...                     the identity on this side.

 

The goal is to use row operations (like you did with Gaussian elimination) to...

[ row 1: 2 , 3 row 2: -4 , -5 | row 1: 1 , 0 row 2: 0 , 1 ] ... turn the left half into I ... and, in the process, the right half with turn into A^( -1 )