 # Let A and B be similar nxn matrices. Prove that if A is idempotent, then B is idempotent. (X is idempotent if X^2=X ) Anish Buchanan 2021-03-18 Answered

Let A and B be similar $n×n$ matrices. Prove that if A is idempotent, then B is idempotent. (X is idempotent if ${X}^{2}=X$ )

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Step 1
According to the given information, it Let A and B be $n×n$ similar matrices.
If A is idempotent the show that B is idempotent.

Step 2
A and B are similar matrices so, there exist an invertible matrix P such that: $B={P}^{-1}AP...\left(A\right)$
Step 3
Square both sides:
${B}^{2}=\left({P}^{-1}AP\right)\left({P}^{-1}AP\right)$
${B}^{2}={P}^{-1}A\left(P{P}^{-1}\right)AP\left[P{P}^{-1}=I\right]$
${B}^{2}={P}^{-1}A\left(I\right)AP$
${B}^{2}={P}^{-1}{A}^{2}P$

Therefore, B is also an idempotent matrix.

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