# If A is a matrix, and the system AX=O has a non-trivial solution, show that there is no matrix, B, such that BA=I.

If A is a matrix, and the system AX=O has a non-trivial solution, show that there is no matrix, B, such that BA=I.
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Cameron Wallace
If $V$ is a nontrivial solution and $BA=I$, then
$V=IV=\left(BA\right)V=B\left(AV\right)=BO=O,$
which contradicts $V$ being nontrivial.
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KesseTher12
𝐴 multiplied on the right by 𝑋 gets 0 and multiplied on the left by 𝐵 gets 𝐼. Multiply on both sides and use associativity, that is, you can first multiply on the right by 𝑋, then on the left by 𝐵, you get the same as: first multiply on the left by 𝐵 then multiply on the right by 𝑋.