Question

The problem simply is this:

Evaluate: 12 5 -2
-3 0 1
-5 4 2

These figures are in matrix form. I don't understand what it is I am supposed to be evaluating.

Answer 1

12 5 -2
-3 0 +1
-5 4 +2

I am going to make a wild guess that it is the determinant they are after. It could be other things, such as inverse, but probably determinant.
There are a lot of ways to find the determinant.

Perhaps the safest is to use a rule that works for any size.
Go element by element down the first column
the determinant is:
first element * determinant of matrix remaining without that row and column
- second element * determinant of remaining matrix without that row and column
= third element * determinant of remaining matrix without that row and column
continue + and - until you reach the bottom
So here we have
12 times the det of
0 1
4 2

+3 times the det of
5 -2
4 +2

+5 times the det of
5 -2
0 +1
Now I will do that first one,
0 1
4 2
using our rule it is 0 (det of remaining matrix which is 2) - 4 (det of remaing matrix which is 1)
= 0*2 - 4*1 = -4
so
we have 12*-4 so far =-48
now we need the second sub matrix determinant to multiply by +3
5 -2
4 +2 Do it yourself and get 18 so for the second prow we get 3*18 = 54
for the third submatrix I got det = 5 and -5 *5 = -25
So , in the end I have
-48 +54 -25 = -19 <--- answer
Now there are easier ways to do a three by three square matrix, but I am playing it safe so you can do other types.

Answer 2

the determinant is:
first element * determinant of matrix remaining without that row and column
- second element * determinant of remaining matrix without that row and column
+ third element * determinant of remaining matrix without that row and column
continue + and - until you reach the bottom

Answer 3

-5 times the det of
5 -2
0 +1