Taking signs out of columns of matrix

Question

My textbook presents the following matrix and asks to find its determinant:

$$d=\left|\begin{array} --1&1&-2\\-1&-1&-4\\-1&1&-7\end{array}\right|$$

And then says that it takes the negative sign from the first column and the negative sign from the third column so that the matrix becomes:

$$d=\left|\begin{array}11&1&2\\1&-1&4\\1&1&7\end{array}\right|$$

I understand that it is possible to multiply any row by -$1$, but if I were to multiply every row by $-1$ then the second column would have the signs inverted. Any hints on how this is possible to take the negative signs out of the matrix?

Answer 1

Rather than premultiplying the matrix, try post multiplying the matrix by $diag(-1, 1, -1)$ to achieve that result.

$$\begin{bmatrix}-1 & 1&-2\\-1&-1&-4\\-1&1&-7\end{bmatrix}\begin{bmatrix}-1 & 0&0 \\ 0&1&0\\0&0& - 1\end{bmatrix}= \begin{bmatrix}1&1&2\\1&-1&4\\1&1&7\end{bmatrix}$$

In terms of determinant, we can perform elementary column operations. You can view it as taking transpose (transpose doesn't change the determiant), perform row operations, then taking transpose back again.

Answer 2

Since $det(A^T) = det (A)$, you can apply operations to columns in just the same way as you apply them to rows, so just multiply first and third columns by $-1$