Thread: Cofactors, adj, 2x2 matrix

I really need some clarification about cofactors and adj of a 2x2 matrix

say the matrix is

a b
c d

what would the cofactors be?

what would the adj be?

this is what i think but for some reason i think its different.

cofactor
d -c
-b a


adj
d -b
-c a


thanks for any clarification about this. i would really appreciate it cause my professor decided to skip this part of the chapter yet its on my online hw

Thank you

You had the right answer!

But as explanation, lets say A is your original matrix:



The definitions are best explained backwards:
. the adjunct adj(A) equals the transpose of the cofactor matrix C
. the transpose of a matrix is one with the rows and columns flipped
. The cofactor matrix C is the matrix of minors M, with each position multiplied by its respective sign
. For a 2 x 2 matrix the minor of each position M[r,c] is the diagonally opposite element.

Using these definitions and starting with A, first create the matrix of minors:



Then the cofactor matrix C has each element of M multiplied by its sign where r,c are the row and column number of each position.



Finally so:



The calculation of adj for 3x3 and larger matrices is the same except that the calculation of the minor matrix requires calculating determinants.

Thank you very much. that was a great explanation