How to find matrix determinant 3 orders

How to find matrix determinant 3 orders

Matrixes exist for display and the solution of systems of the linear equations. One of steps in an algorithm of search of the decision is finding of determinant, or determinant. A matrix 3 orders are a square matrix dimension 3х3.


1. Diagonal from the left top element to right lower is called the main diagonal of a square matrix. From the right top element to the lower left – collateral. The matrix 3 orders has an appearance: a11 a12 a13a21 a22 a23a31 a32 a33

2. For finding of determinant of a matrix of the third order there is an accurate algorithm. At first summarize elements of the main diagonal: a11+a22+a33. Then – the lower left a31 element with average elements of the first line and the third column: a31+a12+a23 (the triangle visually turns out). One more triangle – the right top a13 element and median elements of the third line and the first column: a13+a21+a32. All data composed will pass into a determinant with the sign "plus".

3. Now it is possible to pass to composed with the sign "minus". First, it is collateral diagonal: a13+a22+a31. Secondly, two triangles: a11+a23+a32 and a33+a12+a21. The final formula for search of determinant looks so: Δ=a11+a22+a33+a31+a12+a23+a13+a21+a32-(a13+a22+a31) (-a11+a23+a32) (-a33+a12+a21). The formula is quite bulky, but after some time of practice it becomes habitual and "works" on the automatic machine.

4. In some cases it is easy to see at once that the determinant of a matrix is equal to zero. A zero determinant if any two lines or two columns coincide, are proportional or linearly dependent. If at least one of lines or one of columns completely consists of zero, the determinant of all matrix is equal to zero.

5. Sometimes, to find matrix determinant, it is more convenient and simpler to use transformations of matrixes: algebraic addition of lines and columns among themselves, removal of the general multiplier of a line (column) for the sign of a determinant, a domnozheniye of all elements of a line or a column on the same number. For transformation of matrixes it is important to know their main properties.

Author: «MirrorInfo» Dream Team