Question

Assume A and B are two nonsingular square matrices. Prove that AB has the same eigenvalue as BA.

Answer 1

Step 1:

The Characteristic Polynomial P_AB of AB is given by:

(1)

Step 2:

Equation can be written as follows:

(2)

Step 3:

Equation (2) can be written as follows:

(3)

Step 4:

Equation (3) can be written as follows:

(4)

Step 5:

Equation (4) can be written as follows:

(5)

Step 6:

From equation (5), we get:

(6)

Step 7:
From equation (6), we note that AB and BA have the same characteristic equation.

Given:

A and B are non-singular square matrices. So, A and B are invertible n X n matrices.

So,

by Theorem:

AB has the same eigenvalue as BA.