Question

Suppose that det(a) = a b c d e f g h i = 2 and find the determinant of the given matrix. a b c −4d −4e −4f a + g b + h c + i

Answer 1

I'll go out on a limb and suppose you're given the matrix

$\mathbf A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$

and you're asked to find the determinant of $\mathbf B$, where

$\mathbf B=\begin{bmatrix}a&b&c\\-4d&-4e&-4f\\a+g&b+h&c+i\end{bmatrix}$

and given that $\det\mathbf A=2$.

There are two properties of the determinant that come into play here:

(1) Whenever a single row/column is scaled by a constant $k$, then the determinant of the matrix is scaled by that same constant;

(2) Adding/subtracting rows does not change the value of the determinant.

Taken together, we have that

$\det\mathbf B=-4\det\mathbf A=-8$