Question-39

Select all true statements.

• The number of elements that can be chosen independently in a symmetric matrix of order \(\displaystyle n\) is \(\displaystyle n( n+1) /2\).
• The number of elements that can be chosen independently in a symmetric matrix of order \(\displaystyle n\) is \(\displaystyle n( n-1) /2\).
• The number of elements that can be chosen independently in a skew-symmetric matrix of order \(\displaystyle n\) is \(\displaystyle n( n+1) /2\).
• The number of elements that can be chosen independently in a skew-symmetric matrix of order \(\displaystyle n\) is \(\displaystyle n( n-1) /2\).

Answer

• The number of elements that can be chosen independently in a symmetric matrix of order \(\displaystyle n\) is \(\displaystyle n( n+1) /2\).
• The number of elements that can be chosen independently in a symmetric matrix of order \(\displaystyle n\) is \(\displaystyle n( n-1) /2\).
• The number of elements that can be chosen independently in a skew-symmetric matrix of order \(\displaystyle n\) is \(\displaystyle n( n+1) /2\).
• The number of elements that can be chosen independently in a skew-symmetric matrix of order \(\displaystyle n\) is \(\displaystyle n( n-1) /2\).

Solution

• For a symmetric matrix, the first row has \(\displaystyle n\) free slots, the second row has \(\displaystyle n-1\) free slots and so on. This results in \(\displaystyle n+( n-1) +\cdots +1=\frac{n( n+1)}{2}\) free slots.
• For a skew-symmetric matrix, all diagonal entries are zero. Hence, we need to subtract \(\displaystyle n\) from the above quantity, which results in \(\displaystyle \frac{n( n-1)}{2}\) free slots.