Finding the determinant of this matrix B, using the determinant of Matrix A.

Question

We are provided with a matrix and it's corresponding determine:

This Matrix, let's call it A, has a determinant of 4.

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

We want to find the determinant of this Matrix, B, using A.

\begin{bmatrix} 2c & b & a \\ 2f & e & d \\ 2i & h & g \\ \end{bmatrix}

Now I recognize that B is actually Matrix A, except Column 1 and 3 are swapped and Column 1 in B is multiplied by 2. However, I'm not sure how we can use this information to find it's determinant.

Answer 1

Hint:

the determinant of a matrix is an alternating multilinear form of the columns.

( also see: Defining the determinant of linear transformations as multilinear alternating form)

Answer 2

One can transform the two column operations into a single matrix product:

$$\begin{bmatrix} 2c & b & a \\ 2f & e & d \\ 2i & h & g \end{bmatrix} =\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 2 & 0 & 0 \end{bmatrix} $$ Then it suffices to take the determinants on both sides: you will obtain $4 \times (-2)=-8$.