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Prove that if matrix $C = I - 2M $ and $M = M^2, $ then $ C^3 = C.$

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0

1

$begingroup$

I'm faced with a problem for which I haven't been given a correction, so I expected you could tell me if I am right or not, and in the later case give me the appropriate answer.

Let $M$ be an idempotent matrix such that $M=M^2$, and let $C = I - 2M$, with $I$ being the identity matrix.

Prove that $C^3 = C$

My answer:

$$C=I-2M$$
$$Rightarrow C^2=(I-2M)^2$$
$$=I^2-4IM+4M^2$$
$$=I^2-4M^2+4M^2$$
$$=I$$
$$Rightarrow C^3 = IC=C$$

Thank you.

linear-algebra matrices

share|cite|improve this question

edited Mar 11 at 20:15

J. W. Tanner

3,3351320

asked Mar 11 at 20:05

torito verdejotorito verdejo

745

$endgroup$

• 
  
  
  

  
  6
  

  
  
  
  
  
  

  
  $begingroup$
  In my opinion the idea of your proof is correct. However, you can't use the symbol $iff$. Note that $C=I-2M$ is not equivalent to $C^2=I$. If you use $implies$ it is correct. The same happens in $C^2=Iiff C^3=C.$
  $endgroup$
  – mfl
  Mar 11 at 20:10
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I agree with the above. Also, to be clear, it might be helpful to write $C^3=C^2C=IC=C$
  $endgroup$
  – J. W. Tanner
  Mar 11 at 20:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I changed the implication according to your correction. Thank you.
  $endgroup$
  – torito verdejo
  Mar 11 at 20:17
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Maybe it's worth noting that $(I-2M)^2 = I^2-2IM-2MI+4M^2$ and $IM=MI=M$.
  $endgroup$
  – Wolfgang Kais
  Mar 11 at 20:36
  

  
  
  
  

add a comment | 

0

1

$begingroup$

I'm faced with a problem for which I haven't been given a correction, so I expected you could tell me if I am right or not, and in the later case give me the appropriate answer.

Let $M$ be an idempotent matrix such that $M=M^2$, and let $C = I - 2M$, with $I$ being the identity matrix.

Prove that $C^3 = C$

My answer:

$$C=I-2M$$
$$Rightarrow C^2=(I-2M)^2$$
$$=I^2-4IM+4M^2$$
$$=I^2-4M^2+4M^2$$
$$=I$$
$$Rightarrow C^3 = IC=C$$

Thank you.

linear-algebra matrices

share|cite|improve this question

edited Mar 11 at 20:15

J. W. Tanner

3,3351320

asked Mar 11 at 20:05

torito verdejotorito verdejo

745

$endgroup$

• 
  
  
  

  
  6
  

  
  
  
  
  
  

  
  $begingroup$
  In my opinion the idea of your proof is correct. However, you can't use the symbol $iff$. Note that $C=I-2M$ is not equivalent to $C^2=I$. If you use $implies$ it is correct. The same happens in $C^2=Iiff C^3=C.$
  $endgroup$
  – mfl
  Mar 11 at 20:10
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I agree with the above. Also, to be clear, it might be helpful to write $C^3=C^2C=IC=C$
  $endgroup$
  – J. W. Tanner
  Mar 11 at 20:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I changed the implication according to your correction. Thank you.
  $endgroup$
  – torito verdejo
  Mar 11 at 20:17
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Maybe it's worth noting that $(I-2M)^2 = I^2-2IM-2MI+4M^2$ and $IM=MI=M$.
  $endgroup$
  – Wolfgang Kais
  Mar 11 at 20:36
  

  
  
  
  

add a comment | 

$begingroup$

I'm faced with a problem for which I haven't been given a correction, so I expected you could tell me if I am right or not, and in the later case give me the appropriate answer.

Let $M$ be an idempotent matrix such that $M=M^2$, and let $C = I - 2M$, with $I$ being the identity matrix.

Prove that $C^3 = C$

My answer:

$$C=I-2M$$
$$Rightarrow C^2=(I-2M)^2$$
$$=I^2-4IM+4M^2$$
$$=I^2-4M^2+4M^2$$
$$=I$$
$$Rightarrow C^3 = IC=C$$

Thank you.

linear-algebra matrices

share|cite|improve this question

edited Mar 11 at 20:15

J. W. Tanner

3,3351320

asked Mar 11 at 20:05

torito verdejotorito verdejo

745

$endgroup$

share|cite|improve this question

edited Mar 11 at 20:15

J. W. Tanner

3,3351320

asked Mar 11 at 20:05

torito verdejotorito verdejo

745

share|cite|improve this question

edited Mar 11 at 20:15

J. W. Tanner

3,3351320

asked Mar 11 at 20:05

torito verdejotorito verdejo

745

edited Mar 11 at 20:15

J. W. Tanner

3,3351320

edited Mar 11 at 20:15

J. W. Tanner

3,3351320

asked Mar 11 at 20:05

torito verdejotorito verdejo

745

asked Mar 11 at 20:05

torito verdejotorito verdejo

745