# 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 )

Question
Matrices
Let A and B be similar nxn matrices. Prove that if A is idempotent, then B is idempotent. (X is idempotent if $$X^2=X$$ )

2021-03-19
Step 1
According to the given information, it Let A and B be $$n \times n$$ similar matrices.
If A is idempotent the show that B is idempotent.
$$A \text{ is idempotent } \Rightarrow A^2=A$$
$$\text{ to show } B \text{ is idempotent } B^2=B$$
Step 2
A and B are similar matrices so, there exist an invertible matrix P such that: $$B=P^{-1}AP ...(A)$$
Step 3
Square both sides:
$$B^2=(P^{-1}AP)(P^{-1}AP)$$
$$B^2=P^{-1}A(PP^{-1})AP\left[PP^{-1}=I\right]$$
$$B^2=P^{-1}A(I)AP$$
$$B^2=P^{-1}A^2P$$
$$B^2=P^{-1}AP \left[ \text{by given condition } A^2=A \right]$$
$$B^2=B \left[ \text{from equation } (A) \right]$$
Therefore, B is also an idempotent matrix.

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