Let $n \in \mathbb{N}.$
True or false? Switching two columns of $A \in \mathbb{R}^{n \times n}$ doesn't change $det(A)$. State why.
Let $A=\begin{pmatrix} 1 & 1 & 1\\ 1 & 0 & 0\\ 1 & 0 & 1 \end{pmatrix}$, then $det(A)=-1$
Now switching first column with second column, we have $A'=\begin{pmatrix} 1 & 1 & 1\\ 0 & 1 & 0\\ 0 & 1 & 1 \end{pmatrix}$, then $det(A')=1$
In this example, we saw that the determinant changed, thus the statement is wrong.
Did I do it correctly and is the reasoning correct? Is there a better, more general / valid reasoning?