Determinants

Determinants, in mathematical analysis, a symbolical method by which, inter alia, the solution of equations becomes a matter of mere inspection. The development of the method is due mainly to the French mathematician Cauchy (q.v.). The determinant is the sum of all the products that can be formed from a group of quantities arranged as columns and rows in a square block; each product containing as a factor one from each horizontal row and one from each column; the sign being plus or minus, according as the arrangement of rows from which it is taken requires an even or odd number of transpositions to reduce it to the arrangement in the square. The deter- minant of the third order is equal to $a\; b\text{'}\; c\text{'}\text{'}\; -\; a\; b\text{'}\text{'}\; c\text{'}\; +\; a\text{'}\; b\text{'}\text{'}\; c\; -\; a\text{'}\; b\; c\text{'}\text{'}\; +\; a\text{'}\text{'}\; b\; c\text{'}\; -\; a\text{'}\text{'}\; b\text{'}\; c$.