1. ## 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?

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

cofactor
d -c
-b a

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

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

$A = \begin{array}{lll} a\ \ b \\ c\ \ d \end{array}$

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 $-1^{r+c}$
. 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:

$M = \begin{array}{lll} d\ \ c \\ b\ \ a \end{array}$

Then the cofactor matrix C has each element of M multiplied by its sign $-1^{r+c}$ where r,c are the row and column number of each position.

$C = \begin{array}{lll} \ 1 \times d\ \ -1 \times c \\
-1 \times b\ \ \ 1 \times a \end{array} = \begin{array}{lll} \ d\ \ -c\\ -b\ \ \ a \end{array}$

Finally $adj(A) = C^T$ so:

$adj(A) = C^T = \begin{array}{lll} \ d\ \ -b\\ -c\ \ \ \ a \end{array}$

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

3. Thank you very much. that was a great explanation

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