The diagonal from the upper left element to bottom right called the main diagonal of a square matrix. From the right upper element to the lower-left – side. The matrix is 3 orders of magnitude has the form:a11 a12 a22 a13a21 a23a31 a32 a33
To find the determinant of the matrix of the third order there is a clear algorithm. First, summarize the main diagonal elements: a11+a22+a33. Then, the lower left element of the a31 with moderate elements of the first row and third column: a31+a12+a23 (visually turns the triangle). Another triangle – top-right element a13 and the median of the elements of the third row and first column: a13+a21+a32. All the data components go into the determinant with the plus sign.
Now you can go to term with the sign "minus". First, it's a side diagonal: a13+a22+a31. Second, the two triangles: a11+a23+a32 and a33+a12+a21. The ultimate formula for finding the determinant is: Δ=a11+a22+a33+a31+a12+a23+a13+a21+a32-(a13+a22+a31)-(a11+a23+a32)-(a33+a12+a21). The formula is pretty bulky, but after some time of practice it becomes familiar and "work" on the machine.
In some cases, it is easy to see that the determinant of the matrix equal to zero. The determinant is zero if any two rows or two columns the same, is proportional to or linearly dependent. If at least one row or one column consists entirely of zeros, the determinant of the entire matrix equals zero.
Sometimes to find the determinant of a matrix is more convenient and easier to use matrix transformations: algebraic addition of rows and columns among themselves, making the total multiplier of the row (column) for the sign of the determinant, domlounge all elements of a row or column to the same number. For the transformation matrices is important to know their basic properties.
For the calculation of the determinant there are many specific methods, but, as a rule, in the case of matrices of the third order to apply them impractical.