what are cofactor with its example
In linear algebra, the cofactor of an element in a matrix is a scalar value associated with that element. It is computed by taking the determinant of the matrix formed by removing the row and column containing that element, and then multiplying it by a sign factor (-1) raised to the power of the sum of row index and column index.
Here is an example to clarify:
Consider the following 3x3 matrix:
A = [1 2 3; 4 5 6; 7 8 9]
To find the cofactor of element a_ij, we need to take the determinant of the matrix formed by removing the row i and column j, and then multiply it by (-1)^(i+j).
For example, the cofactor of element a_11 would be:
Cofactor(a_11) = (-1)^(1+1) * determinant of [5 6; 8 9] = (-1)^2 * (5*9 - 6*8) = -3
Similarly, the cofactor of element a_23 would be:
Cofactor(a_23) = (-1)^(2+3) * determinant of [1 2; 7 8] = (-1)^5 * (1*8 - 2*7) = 6
Thus, the cofactors of the elements of matrix A are:
Cofactor(a_11) = -3
Cofactor(a_12) = 6
Cofactor(a_13) = -3
Cofactor(a_21) = 6
Cofactor(a_22) = -12
Cofactor(a_23) = 6
Cofactor(a_31) = -3
Cofactor(a_32) = 6
Cofactor(a_33) = -3