Why is determinant called determinant?

Question

Can someone explain to me why do we call the determinant of a matrix "determinant"? Does it have any meaning? Like it determines something for example!

Answer 1

Here is some information about the origin of the term determinant. This term was introduced the first time $1801$ by C.F. Gauss in his Disquisitiones arithmeticae, XV, p. 2 in connection with a form of second degree.

• The following is from The Theory of Determinants in the historical order of development (1905) by Thomas Muir.

  [Muir, p. 64]: Gauss writes the form as \begin{align*} axx+2bxy+cyy \end{align*} and for shortness speaks of it as the form $(a,b,c)$.

  The function of the coefficients $a,b,c$, which was found by Lagrange to be of notable importance in the discussion of the form, Gauss calls the determinant of the form, the exact words being

• [Gauss, 1801] Numerum $bb-ac$, a cuius indole preprietates formae $(a,b,c)$ imprimis pendere in sequentibus decebimus, determinantem huius formae uocabimus.

and Muir continues:

• [Muir, p.64] ... Here then we have the first use of the term which with an extended signification has in our day come to be so familiar. It must be carefully noted that the more general functions, to which the name came afterwards to be given, also repeatedly occur in the course of Gauss' work, ...

Answer 2

It determines whether a linear system of equations has a solution.

A linear system of $m$ equations in $n$ variables, denoted $A\mathbf{x}=\mathbf{b}$, has a unique solution if the rank of its augmented matrix equals $n$, denoted $\operatorname{rank}(A^{\#}) = n$. (The rank of a matrix is the number of nonzero rows in its row-echelon form and the augmented matrix, $A^{\#}$, is $A$ extended with $\mathbf{b}$ in the last column.)

For $1\times1$ $A$ ($[a_{11}]$), the determinant is $a_{11}$ and the linear system it represents has a solution if $a_{11} \ne 0$ (this is the trivial case of $a_{11}x = b$).

For $2\times2$ $A$, represented

$$ A=\left[ \matrix{ a_{11} & a_{12} \\ a_{21} & a_{22} } \right] $$

the row-echelon form is

$$ \left[ \matrix{ a_{11} & a_{12} \\ 0 & a_{22} - \frac{a_{21}a_{12}}{a_{11}} } \right] $$

so as long as $a_{22} - \frac{a_{21}a_{12}}{a_{11}} \ne 0$, or equivalently $a_{11}a_{22} - a_{12}a_{21} \ne 0$, the system has a unique solution.

Extending this to larger 2D matrices,

$$ \det(A) = \sum \sigma \left( p_1, p_2, \ldots, p_n \right)a_{1 p_1} a_{2 p_2} \cdots a_{n p_n} $$

where the summation is over the $n!$ distinct permutations $\left( p_1, p_2, \ldots, p_n \right)$ of the integers $1, 2, 3, \ldots, n$ and

$$ \sigma \left( p_1, p_2, \ldots, p_n \right) = \begin{cases} +1 & \text{if } \left( p_1, p_2, \ldots, p_n \right) \text{ has even parity}, \\ -1 & \text{if } \left( p_1, p_2, \ldots, p_n \right) \text{ has odd parity.} \end{cases} $$

This is explained well in Chapter 3 of "Linear Analysis and Differential Equations" by Goode and Annin (Amazon Link).

Determinants can be used to calculate areas and volumes in a geometric sense, but the term itself originated from its use in determining whether or not systems of equations have solutions.

NOTE: NOT to be confused with the discriminant, which "discriminates" (i.e. deciphers) the types and numbers of solutions to a polynomial equation.