# Characteristic Polynomial $AB =$ characteristic polynomial $BA$? [duplicate]

Let $$A,B$$ matrix on $$\mathbb{R}$$ size $$nxn$$. How can I prove that $$det(xI - AB) = det(xI - BA)$$ if $$A$$ and $$B$$ are singular matrix

• I take it you know how to prove it when the matrices are nonsingular. You can then get the singular case by continuity. – Gerry Myerson Apr 8 '19 at 0:16
• – Gerry Myerson Apr 8 '19 at 0:21