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?
Best Answer
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.