One mistake students sometimes make is the so-called `proof by example'. The proof by example refers to the situation where the statement is validated for some examples or cases rather than validating for the general case.
Choose the best solution for the question below.
Show that if is a square matrix, then is symmetric.

Question

One mistake students sometimes make is the so-called `proof by example'. The proof by example refers to the situation where the statement is validated for some examples or cases rather than validating for the general case.  
 Choose the best solution for the question below. 
 Show that if    is a square matrix, then    is symmetric.

Answer

Using properties of transposes, we have   . This  implies that    is symmetric.

Answer

For the case where    is a     matrix,  let   . We have   , which is symmetric. 
 For the case where    is a    matrix, let 
  
  We have 
  
 which is symmetric. 
   
 Similarly, we can prove that the statement is true for the   ,    cases, etc. 
 Therefore,  if    is a square matrix, then    is symmetric.

Answer

Taking   , we have 
  
 which is symmetric. 
   
 Taking   ,    is also symmetric because 
  
 Therefore,    is always symmetric.

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One mistake students sometimes make is the so-called `proof by example'. The pro...

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