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 X2=X )

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stuth1
Answered 2021-03-19 Author has 97 answers

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.
A is idempotent A2=A
 to show B is idempotent B2=B
Step 2
A and B are similar matrices so, there exist an invertible matrix P such that: B=P1AP...(A)
Step 3
Square both sides:
B2=(P1AP)(P1AP)
B2=P1A(PP1)AP[PP1=I]
B2=P1A(I)AP
B2=P1A2P
B2=P1AP[by given condition A2=A]
B2=B[from equation (A)]
Therefore, B is also an idempotent matrix.

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Jeffrey Jordon
Answered 2022-01-22 Author has 2047 answers

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