The determinant (or determinant) matrixes is the most important numerical characteristic of a square matrix. Calculation of a determinant of a matrix of the second and third order comes down to application of the simplest formulas. When finding a determinant the matrixes of higher order will require laborious calculations or use of special programs or online services.
It is required to you
- - calculator;
- - handle;
- - paper;
- - computer.
Instruction
1. To find a determinant of a matrix of the first and second order, use the following rules: For a matrix of the first order: ∆1 = a11, For a matrix of the second order: ∆2 = a11*a22 – a12*a21, where: ∆ - the standard designation of a determinant, aij is designation of the element of a matrix located in i-y to a line and in j-m a column.
2. To remember a formula for calculation of a determinant of a matrix of 2х2 in size, use the following formulation: It is necessary to subtract the work of elements of collateral diagonal from the work of the elements located on the main diagonal (passing from top to down, from left to right) (from top to down, from right to left).
3. To find a determinant for a matrix 3х3 choose in it any line or a column – preferably as such in most of which of all zero meet. Then increase each element of this line (column) by a determinant of the matrix 2х2 received by deletion of a line and column, containing this element. Then, the received works need to be put. And, composed, corresponding to odd elements of a line (column) it is necessary to take with a plus, and belonging to even – with a minus sign. The matrix received by deletion of i-y of a line and j-go of a column is called an additional minor (Mij) to the aij element of the main matrix.
4. Example. If for calculation of a determinant to choose the first line of a matrix 3х3, then the aforesaid the rule will turn into the following formula: ∆3 = a11*a22*a33 – a11*a23*a32 – a12*a21*a33 + a12*a23*a31 + a13*a21*a32 – a13*a22*a31
5. The same way arrive if it is required to find a determinant of a matrix of bigger dimension. Only additional minors for a matrix dimension, for example, 4х4 will already have size 3х3 for which calculation of a determinant it is necessary to allocate minors smaller about (2х2).
6. Apparently, with increase in dimension, the complexity of calculations of a determinant of a matrix grows very quickly. On scientific, the number of the elementary calculations necessary for calculation of determinant of a matrix of n x n is designated as About (n!) – i.e. it is comparable with number n! (it is even more than notorious geometrical progression). Already when calculating a determinant for a matrix 4х4 mistake probability is very high therefore for finding of determinants for ""big"" matrixes use online services and applications calculators.