## Characteristic Equation

The equation which is solved to find a Matrix's Eigenvalues, also called the Characteristic Polynomial. Given a system of equations with Matrix

 (1)

the Matrix Equation is
 (2)

which can be rewritten
 (3)

M can have no Matrix Inverse, since otherwise
 (4)

which contradicts our ability to pick arbitrary and . Therefore, M has no inverse, so its Determinant is 0. This gives the characteristic equation
 (5)

where denotes the Determinant of A. For a general Matrix
 (6)

the characteristic equation is
 (7)