Basic Properties and Formulas of Determinants

Table of contents
- Swap rows and columns
Swap rows and columns
  Let $A^{(i\leftrightarrow j)}$ be a matrix given by swapping the $i$ and $j$ columns of a matrix $A$. Its determinant differs from the original determinant only in sign:
determinant swapped columns
Let $A^{(i\hspace{1mm} \updownarrow \hspace{1mm}j)}$ be a matrix given by swapping the $i$ and $j$ rows of a matrix $A$. Its determinant differs from the original determinant only in sign:
determinant swapped rows

Proof
$| A^{(i\hspace{1mm} \updownarrow \hspace{1mm}j)} | = -|A|$
  Let $A_{kl}$ be the $k$-th row and $l$-th column element of an $n \times n$ matrix of $A$. The determinant of $A$ is
(See "Definition of determinant"). Here, the sign of the permutation is
$$ \tag{1.1} $$ Focusing on the $i$ and $j$ rows, let us write $|A|$ as
Let $A^{(i\hspace{1mm} \updownarrow \hspace{1mm}j)}$ be a matrix given by swapping the $i$ and $j$ rows of $A$, and $A^{(i \hspace{1mm}\updownarrow \hspace{1mm}j)}_{kl}$ be the $k$-th row and $l$-th column element of $A^{(i\hspace{1mm} \updownarrow \hspace{1mm}j)}$. We have
The determinant of $A^{(i \hspace{1mm}\updownarrow \hspace{1mm}j)}$ is
$$ \tag{1.2} $$ Here, the last line only changes the order of multiplication.
  Let $\xi$ be a permutation that swaps $\sigma(i)$ and $\sigma(j)$:
$$ \tag{1.3} $$ For example, if $\sigma$ is
and if $\xi$ swaps $\sigma(1)$ and $\sigma(3)$, then
Using $(1.3)$, $(1.2)$ can be expressed as
$$ \tag{1.3} $$ The permutation $\xi$ is an odd permutation because it swaps only once. Therefore, if $\sigma$ is an even permutation, the composite permutation $\xi \circ \sigma(\cdot)$ is an odd permutation, and if $\sigma$ is an odd permutation, the composite permutation $\xi \circ \sigma(\cdot)$ is an even permutation. We therefore see that
$(1.1)$ and this give
holds for any permutation $\sigma$. Using this, $(1.3)$ can be written as
$$ \tag{1.4} $$ The permutation $\sigma$ is a mapping that only changes the order of the set $ \{1,2,\cdots,n\}. $ For example, if $n=3$, all the permutations $\sigma$ are
If $\xi$ swaps $\sigma(1)$ and $\sigma(3)$, all the composite permutations $\xi \circ \sigma $ are
As seen this example, the composite permutation $\xi \circ \sigma$ is also a mapping that only changes the order. And the set
is identical to the set
Therefore, the summation $\sum_{\sigma \in S_{n}}$ in $(1.3)$ can be replaced with $\sum_{\xi \circ \sigma \in S_{n}}$:
Let $\tau$ be $\xi \circ \sigma$. We have
The right hand side is equal to -$|A|$. So we obtain



$| A^{(i \leftrightarrow j)} | = -|A|$
  First, applying the same discussion as above to the transposed matrix $A^{T}$ , we have
Generally, a matrix transposed after swapping columns is equal to a matrix swapping rows after transpose:
We obtain
Finally, the matrix transpose property $|A^{T}| = |A|$ gives