Algebraic addition – an element of matrix or linear algebra, one of concepts of the higher mathematics along with determinant, a minor and the return matrix. However despite the seeming complexity, it is easy to find algebraic additions.
Instruction
1. The matrix algebra as the section of mathematics, is of great importance for record of mathematical models in more compact form. For example, the concept of determinant of a square matrix is directly connected with finding of the solution of systems of the linear equations which are used in a set of applied tasks, including in economy.
2. The algorithm of finding of algebraic additions of a matrix is closely connected with concepts of a minor and determinant of a matrix. The determinant of a matrix of the second order is calculated on a formula: ∆ = a11 · a22 – a12 · a21.
3. The minor of an element of a matrix of an order of n is a determinant of a matrix of an order (n-1) which turns out by the removal of a line and a column corresponding to a position of this element. For example, a minor of the element of a matrix standing in the second line, the third column: M23 = a11 · a32 – a12 · a31.
4. Algebraic addition of an element of a matrix is a minor of an element with the sign which is in direct dependence on what position the element takes in a matrix. In other words, algebraic addition is equal to a minor if the sum of the line number and a column of an element – the even number, and is opposite to it according to the sign when it number – odd: Aij = (-1) ^(i+j)·Mij.
5. Example. Find algebraic additions for all elements of the set matrix.
6. Decision. Use the given formula for calculation of algebraic additions. Be attentive when determining the sign and record of determinants of a matrix: A11 = M11 = a22 · a33 - a23 · a32 = (0 - 10) =-10; A12 = - M12 = - (a21 · a33 - a23 · a31) = - (3 - 8) = 5; A13 = M13 = a21 · a32 - a22 · a31 = (5 - 0) = 5;
7. A21 = - M21 = - (a12 · a33 - a13 · a32) = - (6 + 15) =-21; A22 = M22 = a11 · a33 - a13 · a31 = (3 + 12) = 15; A23 = - M23 = - (a11 · a32 - a12 · a31) = - (5 - 8) = 3;
8. A31 = M31 = a12 · a23 - a13 · a22 = (4 + 0) = 4; A32 = - M32 = - (a11 · a23 - a13 · a21) = - (2 + 3) =-5; A33 = M33 = a11 · a22 - a12 · a21 = (0 - 2) =-2.