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?
Best Answer
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.