A determinant is a value representing the sums and products of a square matrix. The determinant of matrix A, det A, is denoted as an array of numbers or algebraic quantities, called entries or elements, dispersed in horizontal rows and vertical columns enclosed between single vertical lines.

 

Unlike matrices, determinants are always square, having an equal number of rows and columns, which is called the order of the determinant; thus, only square matrices have determinants.

 

The diagonal traversing a determinant from upper left to lower right is the main diagonal, or principal diagonal; the diagonal from the lower left to the upper right is the secondary diagonal.

 

Every determinant represents a definite numerical value that can be calculated according to specific rules that transform the pattern of numbers into a scalar for use in calculations.

 

The numerical value of an n-th order determinant is the algebraic sum of the n! terms, each being the product of n different entries taken one each from every row and column of the determinant.

 

Determinants of order three and higher are usually evaluated by reduction into determinants of a lower order by Laplace expansion into minors. The minor of a determinant is a sub-determinant obtained by removing an equal number of rows and columns to preserve the square shape of the determinant.