$$A = \begin {bmatrix} a & b & c \\ 4 & 0 & 2 \\ 1 & 1 & 1 \end {bmatrix} \ \ , \ \ \left| \ A \ \right| = 3$$
Knowing only this, how does someone calculate the determinant of things like
$$B =\begin{bmatrix} 2a & 2b & 2c \\ 4 & 4 & 4\\ 2 & 0 & 1\end{bmatrix} \ \ \ \text {or} \ \ \ C = \begin{bmatrix} a & b & c \\ 3a + 4 & 3b & 3c+2\\ a+1 & b+1 & c+1\end{bmatrix} \ \ \ \text{?}$$
I don't need the answer, I just have no idea how to do it.
Hint: use multilinearity of the determinant with respect to lines (or columns) and antisymetry.
details: $$\det B =\det \begin{bmatrix} 2a & 2b & 2c \\ 4 & 4 & 4\\ 2 & 0 & 1\end{bmatrix} = 2\times 4\times \frac 12 \det \begin{bmatrix} a & b & c \\ 1 & 1 & 1\\ 4 & 0 & 2\end{bmatrix} = -4\det A = -12 \\ \det C =\det \begin{bmatrix} a & b & c \\ 3a + 4 & 3b & 3c+2\\ a+1 & b+1 & c+1\end{bmatrix} = \det \begin{bmatrix} a & b & c \\ 4 & 0 & 2\\ 1 &1 & 1\end{bmatrix} =\det A = 3 $$