Proving B is Involutory if A is Idempotent

In summary, the conversation discusses proving that B is involutory if and only if A is idempotent, given that B = 2A - I and using the fact that BA = A and AB = A. The first direction is proven by multiplying (1) with A from the right and from the left, separately, and obtaining the results BA = A, and AB = A, which implies B = I. However, the second direction proves to be more challenging and the conversation asks for help. The solution is to square both sides of B = 2A - I, which proves both directions easily.
  • #1
radou
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Let A and B be square matrices of order n, such that B = 2A - I. (1) One has to proove that B is involutory (B^2 = I) <=> A is idempotent (A^2 = A).

Starting off with direction <= , one multiplies (1) with A from the right and from the left, separately, and obtains the results BA = A, and AB = A, respectively. This implies B = I, and I is an involutory matrix, so that direction is prooved. (I hope, that's why I'm asking.)

The second direction, => , causes trouble and I'd appreciate any help. I tried multiplying (1) with B, but it didn't seem to help. Thanks in advance.
 
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  • #2
BA = A and AB = A does not give B = I. For example, take A=0 (there are other examples too). It is true that B=I if BA = A and AB = A holds for all A, but that isn't the case here.

Just square both sides of B=2A-I. This should give both directions pretty easily.
 
  • #3
StatusX said:
BA = A and AB = A does not give B = I. For example, take A=0 (there are other examples too). It is true that B=I if BA = A and AB = A holds for all A, but that isn't the case here.

Just realized that, thanks.

StatusX said:
Just square both sides of B=2A-I. This should give both directions pretty easily.

Did that, and prooved both directions. Thanks again! :smile:
 

What does it mean for B to be involutory?

Involutory means that applying a mathematical operation twice will result in the original value. In this context, it means that if B is involutory, B*B will equal B.

What does it mean for A to be idempotent?

Idempotent means that applying a mathematical operation multiple times will result in the same value as applying the operation once. In this context, it means that if A is idempotent, A*A will equal A.

What is the relationship between B being involutory and A being idempotent?

The relationship is that if A is idempotent, then B being involutory is a sufficient condition for proving that B*B equals B. This means that if B is involutory, then B*B will equal B regardless of the value of A.

How does one prove that B is involutory if A is idempotent?

To prove this, one would use the definition of idempotency to show that A*A equals A. Then, using this information, one would use the definition of involutary to show that B*B equals B. This would prove that B is involutory if A is idempotent.

What are some real-world applications of proving B is involutory if A is idempotent?

This concept is commonly used in computer programming, specifically in binary operations. For example, in Boolean algebra, proving that the NOT operation is involutory if AND is idempotent allows for simplification of logic expressions and efficient coding. It is also used in areas such as linear algebra, group theory, and cryptography.

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