Cofactor matrix of a 2x2 matrix

Last updated: Jan. 2, 2019
  Let $A$ be a 2x2 matrix defined as
The cofactor matrix of $A$ is
cofactor matrix 2x2

  Proof

  Let $M_{ij}$ be a submatrix given by removing $i$-th row and $j$-th column from $A$,
Each element of the cofactor matrix $\tilde{A}$ is defined as
( Note that the subscript order of $M$ is different from that of $\tilde{A}$ ). The determinant of $M_{ij}$ is
Therefore, each element of $\tilde{A}$ is
and in the form of a matrix,
Example:
  Let $A$ be a 2x2 matrix given as
Each element of $\tilde{A}$ is
Calculator
  Enter a 2x2 matrix and press "Execute" button. Then the cofactor matrix is displayed.
Matrix $A$
1 2
1
2
Cofactor Matrix $\tilde{A}$
1 2
1
2