$AB - BA$ can't equal to Unit matrix

Question

Let $A$ and $B$ be square matrices of order $n$. Show that $AB - BA$ can never be equal to unit matrix.

How to approach above question. Please help.

This does not hold in infinite dimensional settings and is crucial for quantum mechanics. It's a very interesting and surprising phenomenon. — Cameron L. Williams, Aug 07 '17 at 14:20

score 5 · Answer 1 · answered Aug 07 '17 at 14:17

5

Hint: take the trace of $AB-BA$ and compare with the trace of the identity matrix.

answered Aug 07 '17 at 14:17

Reveillark

score 2 · Answer 2 · answered Aug 07 '17 at 14:18

2

I know this one from my Linear Algebra course. hint: Take a look at the trace of the matrix $X = AB - BA$.

answered Aug 07 '17 at 14:18

2 Answers2