Cofactor matrix and determinant question

[fluidistic]

Homework Statement

I have the matrix A as being (-2 3 2; 6 0 3; 4 1 -1) and I'd like to calculate its determinant via calculating its cofactor matrix, even if I know it's much more laborious than just calculating its determinant.
I've calculated its cofactor matrix as being (-3 18 6; 5 -6 14; 9 18 -18). I've checked up the arithmetic twice and even redone all the arithmetic but I always fall over this cofactor matrix so I'm almost sure I didn't make any error.
Now from my notes in order to get the determinant, I must sum up all the entries of A multipled with their correspondant in the cofactor matrix.
That is, the arithmetic should start like : $$(-2)\cdot (-3)+3\cdot 18 + 2\cdot 6$$...
I finally get a result of 216 which is $$3 \times 72$$. While calculating the determinant of A simply, I get 72. I'm sure I'm doing something wrong... and also I don't know if this factor 3 is a pure coincidence.
My question is : is it the right way to calculate the determinant of A, supposing that my cofactor matrix is right?

 

[rock.freak667]

I must sum up all the entries of A multipled with their correspondant in the cofactor matrix.

What exactly do you mean by this?

EDIT: I understand what you did now, what you did was expand along all three rows and then added them up.
You only need to expand along one row. (Doing this will give you 72)

Since you did it for all the rows and then added them up you essentially just added 72+72+72 to give 3x72. That's where the factor of three 3 came in.

That is, the arithmetic should start like : $$(-2)\cdot (-3)+3\cdot 18 + 2\cdot 6$$...

That is where the arithmatic should end, as that is 72.

 

[fluidistic]

What exactly do you mean by this?

EDIT: I understand what you did now, what you did was expand along all three rows and then added them up.
You only need to expand along one row. (Doing this will give you 72)

Since you did it for all the rows and then added them up you essentially just added 72+72+72 to give 3x72. That's where the factor of three 3 came in.




That is where the arithmatic should end, as that is 72.

Ah! Thank you so much! Now I understand better my notes, but it wasn't clear at all.