Swap rows and columns

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

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

Let $\xi$ be a permutation that swaps $\sigma(i)$ and $\sigma(j)$:

● $| A^{(i \leftrightarrow j)} | = -|A|$

First, applying the same discussion as above to the transposed matrix $A^{T}$ , we have

Case two columns are equal

Let
$\mathbf{a}_{i}$
and
$\mathbf{a}_{j}$
$(i \neq j)$
denote the $i$-th and $j$-th column vectors of an $n \times n$ matrix $A$,
respectively,
and denote $A$ as
Let $A^{(i\leftrightarrow j)}$ be a matrix given by swapping the $i$-th and $j$-th columns of $A$.

Case two rows are equal

Let
$\mathbf{a}_{i}$
and
$\mathbf{a}_{j}$
$(i \neq j)$
denote the $i$-th and $j$-th row vectors of an $n \times n$ matrix $A$,
respectively,
and denote $A$ as
Let $A^{(i\hspace{1mm}\updownarrow\hspace{1mm} j)}$ be a matrix given by swapping the $i$-th and $j$-th rows of $A$.

Case if a row is scalar-multiplied

Let $A_{ij}$
be the $i$-th row
and $j$-th column element of an $n \times n$ matrix $A$:
Let $A'_{ij}$ be the $i$-th row and $j$-th column element of $A$. The determinant of $A'$ is

Case if a column is scalar-multiplied

Let $A_{ij}$
be the $i$-th row
and $j$-th column element of an $n \times n$ matrix $A$:
Since the determinant of transpose of $A$ is equal to the determinant of $A$ , we have

Case if a scalar is multiplied

Let $A$ be an $n \times n$ matrix.
The determinant of $A$ multiplied by a scaler $\alpha$
is $\alpha^n$ times the determinant of $A$.
By definition of determinant, the determinant of $A$ is

In the same way, the determinant of $\alpha A$ is

Case if a row vector is sum

Let $A$ be an $n \times n$ matrix,
and $\mathbf{a}_{i}$ be the $i$-th row vector of $A$.
From $(7.1)$ and $(7.2)$, the determinants of $|A|$, $|B|$, and $|C|$ are

Case if a column vector is sum

Let $A$ be an $n \times n$ matrix,
and $\mathbf{a}_{i}$ be the $i$-th column vector of $A$.
By $(8.3)$, we have

Add one row to another

Let $A$ be an $n \times n$ matrix,
and $\mathbf{a}_{i}$ be the $i$-th row vector of $A$.
Let $A'$ be an $n \times n$ matrix given by adding the scalar multiple of $j$-th row vector of $A$ to $A$:
By (7.4), we have

Case if the 1st column is 1000...

Let $A$ be
an $n \times n$
matrix whose elements after the second element in the first column are all $0$:
The determinant of $A$ is

Let us write $(10.3)$ by dividing it into the summation for $\sigma(1) = 1$ and the summation for $\sigma(1)\neq 1$.