Let A , B ∈ M n × n ( R ) such that A B − B A = A .
What is the value of det ( A ) ?
Clarifications:
M n × n ( R ) denotes a matrix with n columns and n rows, where every element is a real number.
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A B − B A = A = > A B = B A + A = > A B = A ( B + I ) = > d e t ( A B ) = d e t ( A ( B + I ) ) = > d e t ( A ) . d e t ( B ) = d e t ( A ) . d e t ( B + I ) = > d e t ( A ) ( d e t ( B ) − d e t ( B + I ) ) = 0 .
It is clear that d e t ( B ) = d e t ( B + I ) .So d e t ( A ) = 0
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Suppose d e t ( A ) = 0 . Then A would be invertible and A B A − 1 − B = I ⇒ t r ( A B A − 1 − B ) = t r I = n
But we can also know that t r ( A B A − 1 − B ) = t r ( A B A − 1 ) − t r B = t r ( A − 1 A B ) = t r B = 0 , which is a contradiction.
Then, d e t ( A ) = 0