Section A.2 Determinants
In this section we recall a definition and some properties of determinant of a square matrix. Throughout this section we assume that \(F\) is a field.Let \(S_n\) (\(n\geq 2\)) be the permutation/symmetric group. There is a group homomorphism \(sgn\colon S_n\to\{1,-1\}\) defined by mapping even permutations to \(1\) and mapping odd permutations to \(-1\text{.}\)
Definition A.2.1. (Determinant of a matrix).
Let \(A=(a_{ij})\in M_n(F)\text{.}\) The determinant of \(A\) is defined to be
\begin{equation*}
\det(A)=\sum_{\sigma\in S_n}sgn(\sigma)\; a_{1\sigma(1)}\cdots a_{i\sigma(i)}\cdots a_{n\sigma(n)}\in F.
\end{equation*}
Properties of the determinant. Assume that \(A=(a_{ij})\in M_n(F)\text{.}\)
Row linearity. Let
\(A_i=(a_{i1},a_{i2},\ldots,a_{in})\) be the
\(i\)-th row of
\(A\text{.}\) For
\(\beta\in F\) we let
\(\beta A_i=(\beta a_{i1},\beta a_{i2},\ldots,\beta a_{in})\text{.}\) For any
\(B\in M_{1\times n}(F)\) and any
\(\beta,\gamma\in F\) and any
\(i\in\{1,2,\ldots,n\}\) we get the following.
\begin{equation*}
\det\begin{pmatrix}A_1\\A_2\\\vdots\\A_{i-1}\\\beta A_i+\gamma B\\A_{i+1}\\\vdots\\A_n\end{pmatrix}=\beta\det\begin{pmatrix}A_1\\A_2\\\vdots\\A_{i-1}\\A_i\\A_{i+1}\\\vdots\\A_n\end{pmatrix}+\gamma\det\begin{pmatrix}A_1\\A_2\\\vdots\\A_{i-1}\\B\\A_{i+1}\\\vdots\\A_n\end{pmatrix}
\end{equation*}
For \(A\in M_n(F)\) and any \(\beta\in F\text{,}\) \(\det(\beta A)=\beta^n\det(A)\text{.}\)
Row rearrangement. Let
\(\sigma\in S_n\) and
\(A^\prime\) be the matrix such that the
\(i\)-th row of
\(A^\prime\) is the
\(\sigma(i)\)-th row of
\(A\text{.}\) Then
\begin{equation*}
\det(A^\prime)=sgn(\sigma)\det(A).
\end{equation*}
Alternating. If any two rows of \(A\in M_n(F)\) are the same then \(\det(A)=0\text{.}\)
Transpose. For any
\(A\in M_n(F)\text{,}\)
\begin{equation*}
\det(A^t)=\det(A).
\end{equation*}
Triangular matrices. If
\(A=(a_{ij})\in M_n(F)\) is an upper triangular (resp., lower triangular) matrix, i.e.,
\(a_{ij}=0\) for
\(i>j\) (resp.,
\(a_{ij}=0\) for
\(i<j\)) then
\begin{equation*}
\det(A)=a_{11}a_{22}\cdots a_{nn}.
\end{equation*}
Block form. Let
\(r\in\{1,2,\ldots,n-1\}\text{.}\) Let
\(B\in M_r(F)\text{,}\) \(C\in M_{r\times n-r}(F)\text{,}\) \(D\in M_{n-r}(F)\text{,}\) and
\(\mathbf{0}\in M_{n-r\times r}(F)\) be the zero matrix. The determinant of
\begin{equation*}
A=\begin{pmatrix}B\amp C\\\mathbf{0}\amp D\end{pmatrix}
\end{equation*}
is given by
\begin{equation*}
\det(A)=\det(B)\cdot\det(D).
\end{equation*}
Similar result is true for lower triangular block matrices.
Multiplicative property. Let
\(A, B\in M_n(F)\text{.}\) We have
\begin{equation*}
\det(AB)=\det(A)\cdot\det(B).
\end{equation*}
