Characteristic Polynomials of AB and BA

Theorem: For any and , we have

Proof:

Corollary 1: Let be the characteristic polynomials of respectively. Then

Corollary 2: If are square matrices of the same size, then .

Corollary 3: for all .

Corollary 4: is invertible if and only if is invertible for all .

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I also wrote a Wolfram code snippet to verify the theorem. To avoid floating point errors, are initialized with random integers between and .

m = 8;
n = 5;
ra = 3;
rb = 4;
v = 6;
A = Dot[RandomInteger[{0, v}, {m, ra}], RandomInteger[{0, v}, {ra, n}]];
B = Dot[RandomInteger[{0, v}, {n, rb}], RandomInteger[{0, v}, {rb, m}]];
CharacteristicPolynomial[Dot[A, B], x]
CharacteristicPolynomial[Dot[B, A], x]