Cofactor Formula

A Cofactor is used to calculate the inverse of the matrix, adjoined. The Cofactor is the number you obtain when you eliminate the row and column of an assigned element in a matrix, which is just a numerical grid in the form of a square or a rectangle.

The cofactor is invariably preceded by a positive (+) or negative (-) sign. Suppose that A be an n x n matrix and Mij be the (n – 1) x (n – 1) matrix we get by removing the ith row and jth column. Then, detMij is referred to as the minor of aij. The cofactor matrix formula Aij of aij is described by:

Aij=(−1)i+jdetMij

Cofactor Expansion Formula             

One way of finding out the determinant of an n×n matrix A is to use the following formula known as the cofactor formula.

Choose any i∈{1,…..,n} Then

det(A)=(−1)i+1Ai,1det(A(i∣1))+(−1)i+2Ai,2 det(A(i∣2))+⋯..+(−1)i+nAi,n det(A(i∣n))

Quite frequently we say that the right-hand side (RHS) is the cofactor expansion of the determinant along row i. (The cofactor expansion formula can thus be proved straight out from the definition of the determinant).

There is as well a formula for expanding along column j:

det(A)=(−1)1+jA1,jdet(A(1∣j))+(−1)2+jA2,jdet(A(2∣j))+⋯..+ (−1)n+jAn,jdet(A(n∣j))

At times, one also uses Ci,j to represent (−1)i+j

det(A(i∣j)). We call Ci,j a cofactor of A.

Thus, the cofactor expansion along row i could be expressed as

det(A)=∑j=1nAi,jCi,j,det(A)=∑j=1nAi,jCi,j, and the cofactor expansion along column j could be written as det(A)=∑i=1nAi,jCi,j

We can also use a determinant calculator using cofactor expansion.

Solved Examples

Example: 

Let 

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Then,

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Thus, the minor of a32 is the determinant of this 2 x 2 matrix.

Because the matrix is triangular, the determinant is the product of the diagonals.

(2) (4) = 8

A32=(−1)3+2(8)=−8