What function does the sign of the determinant fulfill? If swapping two rows of a matrix changes the sign of the determinant, then the sign of the determinant does not give me any information about the solutions of the matrix, since they both represent the same system of equations. I am right?
If so, what is the function of the sign of the determinant?
 A: The initial description of matrices as a way to solve a system of equations is not really accurate. I mean, thats what some people use them for, but to mathematicians, a matrix just represents a linear map. In this case, the sign of the determinant tells us whether this map is orientation preserving (positive) or orientation reversing (negative).
For example, consider the linear map $f$ which takes the unit vectors $e_1=(1,0)$ and $e_2=(0,1)$ to $(0,1)$ and $(-1,0)$ respectively. In other words $f$ rotates counterclockwise by $90$ degrees. This is represented by the matrix
$$M=\begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix}.$$
If you lay your right hand along the square spanned by $e_1$ and $e_2$, you can now just rotate your hand to the new basis. However, consider the map $g$ which fixes $e_1$ and takes $e_2$ to $(0,-1)$. The new matrix is
$$N=\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}.$$
Now the image of your right hand under this matrix gets flipped upside down, and we call this "orientation reversing." Notice that $|N|=-1$ while $|M|=1$.
A: Not all matrices represent systems of equations. Matrix multiplication has numerous applications where the sign of the determinant is relevant. Solving systems of equations in a matrix is basically just notation but matrix multiplication is the real workhorse of linear algebra.
For example if I'm considering how $2 \times 2$ real matrices act on the plane $\mathbb{R}^2$ with respect to some basis then the sign of the determinant will tell me if the matrix performs a reflection or not. There is no system of equations in this context but the sign of the determinant contains useful information.
