Tuesday, February 5

Matrix: Cofactors



Consider a matrix that is a square matrx. That means that the matrx is n X n matrx. The number of rows and the number of columns in such a matrx are equal. Each term of the matrx can then have a cofactor. The position of each of the element of a matrx is described by (i,j). Where ‘i’ is row number (ith row) and j is the column number (jth column). For example consider the matrx below:
[a b c]
[d e f]
[g h l]

Here the position of the element a is (1,1) since it is the first element in the first row and first in the first column. Similarly the position of the element f would be (2,3) since it is the third element in the second row. So row number of f is 2 and column number is 3.

Cofactor of matrix:
To be able to find the cofactors of a matrix, we first need to find the minors of each of the elements. This can be done as follows:
If we take any entry, say ‘a’ in the above matrx and remove the row and the column containing a and keep the other entries in the same order, we get the determinant
|e f|
|h l|

This determinant is called a minor of ‘a’. Thus, removing the column and the row containing a given element of a matrx and keeping the surviving entries as they are, yields a determinant called the minor of the given element.

If we now multiply the minor of the entry in the ith row and jth column by (-1)^(i+j), we get the co-factor of that element. Therefore in the above matrx, the minor of h is
|a c|
|d f|
And multiplying this by (-1)^(3+2) = (-1)^5 = -1, we get the cofactor of the element h. This cofactor, therefore is
(-1) * |a c| = -(af – cd)
           |d f|

Note: The co-factor of an element is precisely the factor by which that element is multiplied in the expansion of the determinant of that matrx.

The co factors of a,b,c,… etc are denoted by Ac, Bc, Cc,… and so on.

If every element in a matrx is replaced by its cofactor, the resulting matrx is called the cofactors matrix. Therefore for our matrx above, the cofactors matrx would look like this:
[Ac Bc Cc]
[Dc Ec Fc]
[Gc Hc Lc]

Where, Ac = +(el-fh), Bc = -(dl-fg), Cc = (dh-eg), Dc = -(bl-ch) and so on.

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