How to calculate the determinant of a matrix

Math matrix is a rectangular array of elements (for example complex or real numbers). Each matrix has a dimension of, and denoted by m*n where m is the number of rows, n is the number of columns. At the intersection of rows and columns are the elements of the given set. Matrices are denoted by capital letters A, B, C, D, etc., or A = (aij), where aij is the element at the intersection of the i – th row and j – th column of the matrix. The matrix is called square if it has the number of rows equal to the number of columns. Now we introduce the notion of the determinant of a square matrix of n – th order.

Instruction

1. Consider the square matrix A = (aij) is any n – th order.
   The minor of the element aij of the matrix A is called the determinant of order n -1, the corresponding matrix obtained from the matrix A by striking out from it the i – th row and j – th column, i.e. the row and column which is the element aij. The minor is indicated by the letter M with coefficients: i – row number j – column number.
   A determinant of order n, the corresponding matrix A is a number denoted by the symbol ?. The determinant is calculated according to the formula presented in the figure, where M is the minor of element a1j.
   
2. Thus, if the matrix A has the second order, i.e. n = 2, the corresponding matrix determinant is equal to ? = detA = a11a22 – a12a21
   
3. If the matrix A is of third order, i.e. n = 3, then the corresponding matrix determinant is equal to ? = detA = a11a22a33 ? a11a23a32 ? a12a21a33 + a12a23a31 + a13a21a32 ? a13a22a31
   
4. Calculation of determinants of order n > 3 can be produced method of lowering the order of the determinant, which is based on the zeroing of all but one of the elements of the determinant using properties of determinants.