Question

Can someone explain to me step by step how the book got to this answer i want to understand the steps....thanks

Find the inverse, if it exists,for the matirx (*I used the underscore to separate the numbers within the matix)

-2_1_3
3_-1_2
-4_2_0

the answer is:

-4_6_5
-8_12_13
2_0_-1

The answer you supplied is really not the Inverse matrix since when you multiply the two you get
6 0 0
0 6 0
0 0 6 so each of the numbers in your answer should be divided by 6

There are several ways to find the inverse depending on your level of math knowledge.

Here is a rather straightforward way

let
│-2_1_3│ │a b c│ │1 0 0 │
│3_-1_2│ │d e f│=│0 1 0 │
│-4_2_0│ │g h i│ │0 0 1 │

from this we get
-2a + d + 3g = 1
3a - d + 2g = 0
-4a + 2d = 0

and
-2b + e + 3h = 0
3b - e + 2h = 1
-4b + 2e = 0

and
-2c + f + 3i = 0
3c - f + 2i = 0
-4c + 2f = 1

this gives you 3 systems of 3 equations in 3 variables to solve which do not appear to be that difficult.
Let me know what you got.
(Also check my math, I did not check it)