Question

if A and B two equivalent matrices then prove rank A = rank B​

Answer 1

Step-by-step explanation:

You have B=Q−1APB=Q−1AP, where PP and QQ are invertible. Hence,

ImB=Im(Q−1A)=Q−1ImA.

Im⁡B=Im⁡(Q−1A)=Q−1Im⁡A.

Hence, if you have a basis of ImAIm⁡A, it gets mapped one-to-one (by Q−1Q−1) to a basis of ImBIm⁡B, which implies

rankB=dimImB=dimImA=rankA

hope it helps