Question

In a third order matrix $A$, $a_{ij}$ denotes the element in the i-th row and j-th column.
If $a_{ij}=0$ for $i=j$
$=1$ for $i>j$
$=-1$ for $i<j$
Then the matrix is

• A. skew symmetric
• B. symmetric
• C. not invertible
• D. non-singular

Answer 1

Answer: (A), (C)

The correct options are
A skew symmetric
C not invertible

$A=\left[\begin{array}{ccc}0& -1& -1\\ 1& 0& -1\\ 1& 1& 0\end{array}\right]$ skew symmetric
$\left|A\right|=\left|\begin{array}{ccc}0& -1& -1\\ 1& 0& -1\\ 1& 1& 0\end{array}\right|=0+(0+1)-1(1-0)=1-1=0$
$\left|A\right|=0$ $\Rightarrow$ non invertible