What does the co-factor of a matrix represent? What does the co-factor of a matrix represent. I am not asking for the formula to calculate co-factors or for their specific usage in matrix operations but rather for the meaning behind co-factors. What are the co-factors of a matrix in physical terms?
 A: A matrix cofactor is associated with some element $i,j $ of the matrix $A $. The cofactor$_{i,j}$ is the determinant of the submatrix that results from taking the row $i $ and column $j $ from $A $, times $(-1)^{i+j} $. As it turns out, finding all the cofactors of $A $ can be helpful to solve linear systems with Cramer's rule and can also be used to invert matrices. Those are its most basic applications.
A: The determinant of a 3x3 matrix is equal to the scalar triple product of its rows, or of its columns.  The scalar triple product has the well-known physical interpretation as the volume of a parallelepiped.
In the 3D case, the cofactors of a row (column) are the components of the cross product of the other two rows (columns).  In this case, the cofactors have the physical interpretation as the area of the parallelogram defined by the other two rows (columns) with one component deleted.  More specifically, the cofactor of element $a_{11}$ is (the magnitude of) the cross product of either the rows $(0,a_{22},a_{23})$ and $(0,a_{32},a_{33})$, which is the same as the cross product of the columns $(0,a_{22},a_{32})$ and $(0,a_{23},a_{33})$.  
In the 4D case, the cofactors are determinants of 3D matrices, so the cofactors are volumes in a 3D space.  The determinant of the 4D matrix is the scalar product of a row (column) with the vector of its cofactors.  
In general, the determinant of an $n$ by $n$ matrix is the volume of a box in an $n$-dimensional space.  Since the cofactors are determinants of $(n-1)$ by $(n-1)$ matrices, they are volumes in a $n-1$ dimensional space.
