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Consider the following statements in respect of a square matrix A and its transpose AT

1. A + AT is always symmetric. 

2. A - AT is always anti-symmetric.

Which of the statements given above is/are correct ?


1. 1 only 
2. 2 only 
3. Both 1 and 2 
4. Neither 1 nor 2
5. None of these

1 Answer

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Best answer
Correct Answer - Option 3 : Both 1 and 2 

Concept:

Consider a matrix A and its transpose AT:

(AT)T = A, (A + B)T = AT + BT

A matrix ‘A’ is said to be symmetric if A = AT and anti-symmetric if A = -AT

Calculation:

We know, that A matrix ‘A’ is said to be symmetric if A = AT and anti-symmetric if A = -AT

Now, consider \(\rm (A+A^T)^T=A^T+(A^T)^T =A^T+A\)

Therefore, A + AT is always symmetric. 

Now, consider \(\rm (A-A^T)^T=A^T-(A^T)^T =A^T-A\)

= -(A - AT)

Therefore, A - AT is always anti-symmetric. 

Hence, option (3) is correct. 

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