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

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  • 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 08 '19 at 00:16
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    See also https://math.stackexchange.com/questions/2092168/are-the-eigenvalues-of-the-matrix-ab-equal-to-the-eigenvalues-of-the-matrix-ba and https://math.stackexchange.com/questions/1787537/help-with-understanding-the-proof-for-ab-and-ba-have-the-same-characteristi and https://math.stackexchange.com/questions/378303/show-ab-and-ba-have-the-same-eigenvalues and https://math.stackexchange.com/questions/232561/eigen-values-proof and https://math.stackexchange.com/questions/2163564/eigenvalues-for-a-product-of-matrices and many others. – Gerry Myerson Apr 08 '19 at 00:21
  • @GerryMyerson He did not specify the field (or whether the coefficients even are in a field). – Igor Rivin Jun 17 '22 at 03:52
  • @Igor, the very first sentence of the question uses the standard notation for the field of real numbers. – Gerry Myerson Jun 17 '22 at 03:55
  • @GerryMyerson True. But the statement is true over any commutative ring, as you know better than I. – Igor Rivin Jun 17 '22 at 04:02

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