66.17: Matrix Properties
( \newcommand{\kernel}{\mathrm{null}\,}\)
This appendix presents a brief summary of the properties of 2×2 and 3×3 matrices.
2×2 Matrices
Determinant
The determinant of a 2×2 matrix is given by the well-known formula:
det(abcd)=ad−bc.
Matrix of Cofactors
The matrix of cofactors is the matrix of signed minors; for a 2×2 matrix, this is
cof(abcd)=(d−c−ba)
Inverse
Finally, the inverse of a matrix is the transpose of the matrix of cofactors divided by the determinant. For a 2×2 matrix,
(abcd)−1=1ad−bc(d−b−ca)
3×3 Matrices
Determinant
The determinant of a 3×3 matrix is given by:
det(abcdefghi)=a(ei−fh)−b(di−fg)+c(dh−eg)
Matrix of Cofactors
The matrix of cofactors is the matrix of signed minors; for a 3×3 matrix, this is
cof(abcdefghi)=(ei−fhfg−didh−egch−biai−cgbg−ahbf−cecd−afae−bd)
Inverse
Finally, the inverse of a matrix is the transpose of the matrix of cofactors divided by the determinant. For a 3×3 matrix,
(abcdefghi)−1=1a(ei−fh)−b(di−fg)+c(dh−eg)(ei−fhch−bibf−cefg−diai−cgcd−afdh−egbg−ahae−bd)