For calculus and mathematics students, working with matrices and determinants is a part of their daily routine. The topic of determinants is generally introduced to the students while teaching them about matrices; the reason is that a determinant can tell us many things about a square matrix. Firstly, a non-zero determinant can accurately predict if the matrix can be successfully inverted. As matrix inversion is a far tedious process, the determinant check is almost always employed first to confirm if a matrix inverse exists.

Determinants are generally computed by adding & subtracting the products of elements of a square matrix in a pre-defined pattern. The absolute value obtained after the calculation can be considered an area or volume, depending on the order of the matrix. However, historically, determinants were defined without any reference to matrices. In the 3rd Century BC, the Chinese considered determinants as an important property of a system of linear equations; useful in determining if a unique solution for the system exists or not. The method of evaluating a determinant to arrive at its absolute value was proposed by the French Mathematician, Laplace, in the 18th century; though he called them “resultant”. In the same century, Lagrange, the famous Italian Mathematician worked on second and third order determinants and applied it solve problems involving elimination theory. Gauss, the German Mathematician and scientist, first used the word “determinants” and applied it to number theory. Gauss also contributed to finding the reciprocal of a determinant.

There are many interesting properties of determinants that make computing them much easier. The most important properties are:

1. A matrix and its transpose, both have the same determinant.
2. The determinant of a triangular matrix is simply the product of the diagonal elements.
3. By interchanging two rows, the determinant of the matrix retains its absolute value, only its sign changes.
4. If we modify our matrix by multiplying one row by a constant, we do not need to re-evaluate the determinant, but just multiply the constant with the old determinant result.
5. If a row has no non-zero elements, the determinant is always zero.
6. Adding one row to another one multiplied by a constant, has no effect on the result of the determinant.

If you work with a lot of determinants, a scientific calculator can easily serve as your determinant calculator. The algorithm for evaluating a determinant is so simple that you can even write a program for your own determinant calculator within half an hour, in C++, BASIC, or any other language of your choice.

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# Determinant Properties

In mathematics, a lot of attention is paid to the solution of matrixes-rectangular tables of numbers having a certain number of rows and columns, and which, accordingly, have different forms and sizes. Any square matrix, then there is one in which the same number of rows and columns that correspond to a specific number, which also is called the determinant or determinant. It should be noted that the determinants exist only for square matrices of the form. Finding the determinant by using the ...

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