Null matrix can be an idempotent matrix?

[gianeshwar]

Friends!
Is there anything wrong in the statement"A null matrix is idempotent".

 

[Mark44]

Friends!
Is there anything wrong in the statement"A null matrix is idempotent".

What do you think? How is the term "idempotent" defined?

 

[gianeshwar]

A square equal A. So if A is zero matrix then it satsfies this condition.

 

[Mark44]

A square equal A. So if A is zero matrix then it satsfies this condition.

Any square zero matrix A is idempotent, since A² = A. So, in regard to your question in the first post, not all zero matrices are idempotent, since A² might not be defined. For example,
$$A = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$$

 

[gianeshwar]

Thank You.Doubt had arisen while showing B square not equal to I to be not always true when given B equal I minus A and A is idempotent.

 

[Mark44]

Thank You.Doubt had arisen while showing B square not equal to I to be not always true when given B equal I minus A and A is idempotent.

I'm not sure what you're trying to say here. Writing an equation would make it clearer.

Are you saying that B = I - A, and A is idempotent? And you need to show that B² ≠ I?

If so, that's easy to show -- just expand B² = (I - A)².

 

[gianeshwar]

I needed to show that( B squarenot equai to I) is not always true.

 

[Mark44]

I needed to show that( B squarenot equai to I) is not always true.

Like I said in my previous post, this is easy to do.

Also, please take some time to learn how to write proper math notation. To indicate the square of something, at the very least you can write B^2. Better yet, write the exponent as a superscript, like this: B². The x² icon in the menu bar can be used to write exponents of all kinds. The $\Sigma$ icon in the menu bar has many math symbols, such as ≠ and ∞ and many others.

 

[gianeshwar]

Thank You Mark44 for the valuable help and suggestions!

 

[Mark44]

Thank You Mark44 for the valuable help and suggestions!

You're welcome!