• For a `ntimes n` matrix `A`, the determinant of `A` denotes the magnification of the volume of `n`-dimensional parallelopiped.
  
• More formally, the following are the determinants of `2times 2` and `3times 3` matrices.

   If `A=(a_1, a_2)` is a `2times 2` matrix, `det A` means the transformed area from the unit area. If `A=(a_1, a_2, a_3)` is a `3times 3` matrix, `det A` means the transformed volume from the unit volume.

• For the arbitrary column vector `a_i` of `A`, adding `ka_i` to another column vector `a_j` does not change `det A`. For example, `det(a_1, a_2, a_3)=det(a_1, a_2, a_3+ca_2)` for `c in mathbb{R}`. Consider `det A` as a deck, then its volume is not changed although it is pushed to some directions.

• The determinant of a upper triangular matrix `A` is the product of all the diagonal elements of `A`.

\begin{align} A= \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{pmatrix} \Rightarrow \det A=a_{11}a_{22}a_{33} \end{align}

• For `A=(a_1, cdots, a_n)` and `c in mathbb{R}`, `quad det(c a_1, a_2, cdots, a_n)=``c det(a_1, cdots, a_n)`. Moreover, `det(a_1+a_1^{prime}, a_2, cdots, c_n)=``det(a_1, a_2, cdots, a_n)+``det(a_1^{prime}, a_2, cdots, a_n)` for an arbitrary `n times 1` vector `a_1^{prime}`.

︎Reference

[1] Hiraoka Kazuyuki, Hori Gen, Programming No Tame No Senkei Daisu, Ohmsha.