Show that if A and B are an n n matrices such that AB =O (where O is the zero matrix) then A is not invertible or B is not invertible.

Question
Matrix transformations
asked 2020-12-03
Show that if A and B are an n n matrices such that AB =O (where O is the zero matrix) then A is not invertible or B is not invertible.

Answers (1)

2020-12-04
If A is not invertible we are done. If A is invertible, we can multiply AB=0 by \(\displaystyle{A}^{{-{{1}}}}\) from the left:
\(\displaystyle{A}^{{-{{1}}}}{\left({A}{B}\right)}={A}^{{-{{1}}}}{O}\to{B}={0}\)
So, B=0, meaning that B is not invertible.
0

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