Question

In a third order matrix $A,a_{ij}$ denotes the element in the $i-\text{th}$ row and $j-\text{th}$ 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

$A=\left[\begin{array}{ccc}0& -1& -1\\ 1& 0& -1\\ 1& 1& 0\end{array}\right]$
$A^T=-A$
$\Rightarrow A$ skew symmetric.

$|A|=\left|\begin{array}{ccc}0& -1& -1\\ 1& 0& -1\\ 1& 1& 0\end{array}\right|$
$=0$

$|A|=0\Rightarrow A$ is singular and non invertible.