Page 150 - Schaum's Outline of Differential Equations

CHAP. 15] MATRICES 133

THE CHARACTERISTIC EQUATION
The characteristic equation of a square matrix A is the polynomial equation in A, given by

where det( ) stands for "the determinant of." Those values of A, which satisfy (15.1), that is, the roots of (15.1),
are the eigenvalues of A, a Mold root being called an eigenvalue of multiplicity k.
Theorem 15.1. (Cayley—Hamilton theorem). Any square matrix satisfies its own characteristic equation.
That is, if

then

Solved Problems

15.1. Show that A + B = B + A for

Since the corresponding elements of the resulting matrices are equal, the desired equality follows.

15.2. Find 3A - |B for the matrices given in Problem 15.1.

15.3. Find AB and BA for the matrices given in Problem 15.1.

Note that for these matrices, AB ^ BA.