11. Evaluation of determinants
In principle. you can compute a determinant with given elements with the aid of the definition (6.1). You can also employ the expansion formulae, which reduce the calculation of an n-th order determinant to that of n minors of order (n - 1); you can then again apply these formulae to the minors, etc. However, this would lead to extensive, inconvenient computations. Hence there is room for certain rules which in many cases considerably simplify the task.
The computation of a 2-nd order determinant is very simple:
you subtract from the product of the elements in the main diagonal the product of the elements in the bottom left to top right diagonal. Also for the calculation of a determinant of order 3, there exists a similar method, the so-called Sarrus Rule. In order to compute
you rewrite the first two columns of A to the write of A. In this matrix with 3 rows and 5 columns, there exist three inclined rows from the top on the left hand side to the bottom on the right hand side, and three rows from the top left to the bottom right, each with three elements. You now form the products of the elements in the inclined rows, give the first group of inclined rows a + sign, the second group of inclined rows a - sign and add these products. A comparison of this expression with that for d3 in Section 5 shows that the result is the determinant |A|. The following pattern illustrates Sarrus' Rule:
A triangular matrix has only zeroes either above or below the main diagonal. It is readily confirmed that the determinant of a triangular matrix is the product of the elements in the main diagonal.
The proof is obtained by induction with respect to the grade n of the matrix. For n = 1, nothing has to be proved. Assume that the statement has been proved for triangular matrices of order less than n. In order to compute the determinant of the triangular matrix
expand |A| with respect to the first row. You find |A| = a11A11, where A11 is the determinant of a triangular matrix of order (n - 1), so that by induction
A11 =a22a33 ···ann.
For triangular matrices with zeroes below the main diagonal, our statement follows from Theorem 4.
The following example demonstrates how to employ the properties of determinants. Compute
You add the first row to the second and third rows and subtract them from the third row:
Expansion with respect to the last column yields
You add the third row to the second row and extract 5 from the second row
You now add the second row to the first row and three times the third row to the second row; finally, you expand with the respect to the last column:
