Question

How do you find the inverse of `A=``((2, -4), (1, 3))`?

Answer 1

`A^(-1)=((3/10,2/5),(-1/10,1/5))`

Explanation:

Extend `A` with an identity matrix appended on its right:

`((2,-4,"|",1,0),(1,3,"|",0,1))`

Perform standard row operations with the objective of modifying the left side into an identity (sub)matrix.
(There are many ways to do this; the following is just one method).

Exchange Rows 1 and 2
`((1,3,"|",0,1),(2,-4,"|",1,0))`

Subtract `2xx` Row 1 from Row 2
`((1,3,"|",0,1),(0,-10,"|",1,-2))`

Divide Row 2 by `(-10)`
`((1,3,"|",0,1),(0,1,"|",-1/10,1/5))`

Subtract `3xx` Row 2 from Row 1
`((1,0,"|",3/10,2/5),(0,1,"|",-1/10,1/5))`

with the identity matrix on the left, the right side is the required inverse.

Of course, it's a good idea to verify this result:
`((2,-4),(1,3))xx((3/10,2/5),(-1/10,1/5))`

`=((6/10+4/10,color(white)("XXX"),4/5-4/5),(3/10-3/10,,2/5+3/5))`

`=((1,color(white)("XX"),0),(0,,1))`
...and we can go home happy!