Question

what is the minor and cofactor

Answer 1

Minor Entries of a Matrix

We have already looking at a method for evaluating the determinant of a 2×2 square matrix A=[acbd], that is det(A)=ad−bc. We will soon look at a method for evaluating the determinants of larger square matrices with what are known as minor entries and cofactors. For the time being, we will need to introduce what minor and cofactor entries are. First we will introduce a new notation for determinants:

(1) det(A)=∣∣∣acbd∣∣∣=ad−bc

Of course this notation is not limited to 2×2 matrices as we will see.

Definition: If A is a square matrix of size n×n, the Minor Entry Mij is the determinant of the (n−1)×(n−1) submatrix after deleting the ith row and jth column from A.

For example, consider the following 3×3 matrix:

(2) A=⎡⎣147258369⎤⎦

The minor entry M11 is the determinant of the 2×2 matrix after deleting the 1st row and the 1st column. Therefore M11=∣∣∣5869∣∣∣. We can calculate this determinant by the rule we used earlier, so M11=(5)(9)−(8)(6)=−3. Therefore the minor entry M11=−3.

If we wanted, we could calculate other minor entries. For example, let's calculate M23, that is the determinant of the submatrix after deleting the second row and third column of A, M23=∣∣∣1728∣∣∣=1(8)−7(2)=−6.

Cofactors Entries of a Matrix Definition: Given that Mij is the minor entry for row i and column j in a square matrix A, the Cofactor of row i and column j is denoted as Cij and is calculated by the formula Cij=(−1)i+jMij.

We note that if the sum i+j is even, then Cij=Mij, and that if the sum is odd, then Cij=−Mij. Hence, the only difference between the related minor entries and cofactors may be a sign change or nothing at all. Whether Cij=Mij or Cij=−Mij has a pattern for square matrices as illustrated:

For example C12=−M12. Of course if you forget, you can always use the formula Cij=(−1)i+jMij, for example C12=(−1)1+2Mij=(−1)3Mij=−Mij.

Now let's try an actual example by finding C31 given the following matrix:

(3) A=⎡⎣3−52369358⎤⎦

After deleting the 3rd row and the 1st column, we find M31 by taking the determinant of this submatrix, that is M31=∣∣∣3635∣∣∣=(3)(5)−(3)(6)=−3 We then note that i+j=3+1=4 which is even, so Mij=Cij=−3, so the cofactor entry of row 3 column 1 is −3.

Answer 2

Minors and Cofactors: Expand with a line. Finder of 2 × 2 matrix is ​​easy to find: You simply multiply cross-cross, and subtract: ......A "minor" is the determinant of the square matrix created by removing one row and one column from a large square matrix .