# 56. Determinants¶

Let $$a, b, c, d$$ be any four numbers, real or complex, the symbol

$\begin{split}\begin{vmatrix} a & b\\ c & d\\ \end{vmatrix}\end{split}$

denotes $$ad - bc$$ and is called a determinant of second order. $$a, b, c, d$$ are called elements of the determinant and $$ad - bc$$ is called value of the determinant.

As you can see, the elements of a determinant are positioned in the form of a square in its designation. The diagonal on which elements $$a$$ and $$d$$ lie is called the principal or primary diagonal of the determinant and the diagonal which is formed on the line of $$b$$ and $$c$$ is called the secondary diagonal.

A row is constituted by elements lying in the same horizontal line and a column is constituted by elements lying in the same vertical line.

Clearly, determinant of second order has two rows and two columns and its value is equal to the products of elements along primary diagonal minus the product of elements along the secondary diagonal. Thus, by definition

$\begin{split}\begin{vmatrix} 2 & 4\\ 3 & 9\\ \end{vmatrix} = 18 - 12 = 6\end{split}$

Let $$a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$$ be any nine numbers, then the symbol

$\begin{split}\begin{vmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\\ \end{vmatrix}\end{split}$

is another way of saying

$\begin{split}a_1\begin{vmatrix} b_2 & b_3\\ c_2 & c_3\\ \end{vmatrix} - a_2\begin{vmatrix} \end{vmatrix}\end{split}$