Algebra rules when multiplying matrices with unknown variables I'm about to compute the determinant of a given matrix, this matrix however contains variables instead of actual values, so I'm a bit uncertain what to do here.
I have the 3x3 matrix A:
$$A = \left(\begin{matrix} a-b-c & 2a & 2a\\ 2b & b-c-a & 2b\\ 2c & 2c &c-a-b\end{matrix}\right).$$
When computing the determinant i start off by taking the first element and multiply this with the determinant of the $2\times 2$ matrix and here I encounter the problem as I'm not sure how multiplying work out in this scenario:
$$\det(A) = (a-b-c)\left((b-c-a)(c-a-b) - (2b)(2c)\right). $$
Do I start off by multiplying $b$ with $c$, then $-a$ and then $-b$ or do I try to simply the expressions, like $b$ cancels out $-b$ etc.?
This is my attempt for the determinant of the $2\times 2$ matrix for the first part:
\begin{eqnarray*} (b-c-a)(c-a-b) &= &bc - ab -b^2 - c^2 + ac + bc - ac + a^2 + ab \\&=& a^2 - b^2 + 2ab - c^2.\end{eqnarray*}
 A: Since the determinant of a matrix is defined in terms of sums of products, which entails the use of the standard operations of addition and multiplication, the "algebra" needed for computing the determinant is simply the standard arithmetic manipulation on variables and scalars.
First, note that you have not obtained the full expression for $\det (A)$ when expanding along the first column.
$$\begin{align} \det(A) & = (a-b-c)\Big((b-c-a)(c-a-b) - (2b)(2c)\Big) \\ 
& - \;\;2b\Big(2a(c-a-b) - (2a)(2c)\Big) \\ 
& + \;\;2c\Big((2a)(2b) - 2a(b-c-a)\Big)\end{align}$$
Also, for the product within the first term of the sum, we have (note the bold-face term, which is in contrast to your computation):
\begin{eqnarray*} (b-c-a)(c-a-b) &= &bc - ab -b^2 - c^2 + ac + bc - ac + a^2 + ab \\&=& a^2 - b^2 + {\bf 2bc} - c^2.\end{eqnarray*}
Then for the complete first term of the sum, we need to compute:
$$(a - b - c)\Big((a^2 - b^2 + 2bc - c^2)- 4bc\Big) = (a - b - c)(a^2 - b^2 -c^2 - 2bc)$$
The second and third term in $\det(A)$ should be a bit less tedious to compute.
