Cofactor matrix of a 3x3 matrix

Last updated: August 5th, 2017
  Let $A$ be a 3x3 matrix given as
3x3 matrix
. The cofactor matrix of $A$ is
The cofactor matrix of a 3x3 matrix

  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
cofactor matrix definition
(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,
3x3 cofactor matrix
Example:
  Let $A$ be a 3x3 matrix given as
Each element of $\tilde{A}$ is
Example of a 3x3 cofactor matrix
Calculator
  Enter a 3x3 matrix and press "Execute" button. Then the cofactor matrix is displayed.
Matrix $A$
1 2 3
1
2
3
Cofactor Matrix $\tilde{A}$
1 2 3
1
2
3