To calculate the determinant of a matrix in two ways: by the method of triangle or expanding it on the elements of the row or column. In the second case this number is obtained by summing the products of three components: the values of the elements themselves, (-1)^k and the matrix of the minors of order n-1:∆ = Σ a_ij•(-1)^k•M_j, where k=i+j – sum of numbers of the element, n is the dimension of the matrix.
The determinant can only be found for square matrices of any order. For example, if it is equal to 1, the determinant is a single element. For the matrix of the second order starts to operate, the above formula. Divide the determinant by the elements of the first row:∆_2 = a11•(-1)2•M11 + a12•(-1)3•M12.
Minor of the matrix is also a matrix, which is 1 less. It is obtained from the source using the algorithm of deleting the corresponding row and column. In this case, the minors will consist of one element, because the matrix has a second dimension. Remove the first row and first column and you will get M11 = a22. Cross out the first row and the second column, and find M12 = a21. Then the formula will be as follows:∆_2 = a11•a22 – a12•a21.
The determinant of the second order is one of the most common in linear algebra, so this formula is used very often and does not require constant removal. Similarly, it is possible to calculate the determinant of the third order, in this case the expression is more cumbersome and consist of three components: elements of the first row and their minors:∆_3 = a11•(-1)2•M11 + a12•(-1)3•M12 + a13•(-1)^4•M13.
It is obvious that the minors of this matrix will be of the second order can therefore be calculated as the determinant of the second order according to the rule given earlier. Consistently deleted: string1+Column1, Line1+and Line1 Column2+column3:∆_3 = a11•(a22•a33 – a23•a32) – a12•(a21•a33 – a23•a31) + a13•(a21•a32 – a22•a31) == a11•a22•a33 +a12•a23•a31 + a13•a21•a32 – a11•a23•a32 – a12•a21•a33 – a13•a22•a31.