Minors and Cofactors of a 2x2 Matrix?

Question

I was absent in class when my teacher taught us this so I would reeaally appreciate if someone could help me out.
The matrix (pretend that there are brackets):
3 4
2 -5

Please tell me how you got to your answer also!

Thanks :)

Answer 1

Minors and co-factors are usually associated with larger matrices. It is a method of dividing the problem of calculating the determinant into a set of smaller tasks, hopefully easier individually that result in the answer to the more complex task.



The determinant of this matrix is 3*(-5)-2*4=-23



A "minor" is the determinant of the square matrix formed by deleting one row and one column from some larger square matrix. Since there are lots of rows and columns in the original matrix, you can make lots of minors from it. These minors are labeled according to the row and column you deleted. So if you were to go, say, to the a2,4 entry from some matrix A and cross out the row and column that pass through that entry (that is, if you remove the second row and the fourth column of the matrix), the determinant of the new (and smaller) matrix is called "the minor M2,4".



There are therefore four potential minors |3|, |4|, |2| and |-5| These would be used in pairs



example:

Minor(1,1) = |-5|

Minor(1,2) = |2|



Co-factor(1,1) = 1^(1+1)*3 = 3

Co-factor(1,2) = -1^(1+2)*4 = -2



The determinant is therefore:

Co-factor(1,1) *Minor(1,1) +Co-factor(1,2) *Minor(1,2) = 3*(-5)+(-2)*4=-23

Answer 2

The cofactor of a_(11) is 7, i.E., the 7 in the cut back proper. You assess cofactors with the aid of disposing of the row and column that a targeted entry lies in. For a_(11) you eliminate the first row and the first column and you might be left with 7. For a_(12), you get rid of the primary row and the 2d column and you are left with 6, however the rule is that if the subscripts sum to an atypical quantity you exchange the signal. The cofactor of a_(12) is -6. If the subscripts sum to a good number you do not alternate the sign. The matrix of cofactors for this reason is [ 7 -6 ] [ -2 3 ] Adj(A) is the transpose of this. By the way, for greater rectangular matrices the cofactor continues to be discovered by way of taking away the proper row and column, but then you take the determinant of what stays.

Answer 3

Now i have appriciated/ you simply find by using the normal method by fixing i-row and j-column and obtaining the determinant of the left matrix