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## Determinant

• 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

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

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