Question

In a skew-symmetric matrix, the diagonal elements are all
A) One
B) Zero
C) Different from each other
D) Non-zero

Answer 1

Hint: A square matrix \[A = \left[ {\mathop a\nolimits_{ij} } \right]\]is said to be skew symmetric matrix if
$A' = - A$ or $A = - A'$, that is \[\mathop a\nolimits_{ij} = - \mathop a\nolimits_{ji} \]for all possible values of $i$ and $j$.
In transpose of a matrix, columns and rows are interchanged. Transpose denoted by: $A'{\text{ (or }}\mathop A\nolimits^T )$. For example:
If \[A = {\left[ {\begin{array}{*{20}{c}}
  3 \\
  {\sqrt 3 } \\
  0
\end{array}{\text{ }}\begin{array}{*{20}{c}}
  5 \\
  1 \\
  {\dfrac{{ - 1}}{5}}
\end{array}} \right]_{3 \times 2}}\]
Then \[A' = {\left[ {\begin{array}{*{20}{c}}
  3 \\
  5
\end{array}\begin{array}{*{20}{c}}
  {\sqrt 3 } \\
  1
\end{array}\begin{array}{*{20}{c}}
  0 \\
  {\dfrac{{ - 1}}{5}}
\end{array}} \right]_{2 \times 3}}\]

Complete step-by-step answer:
Step 1: Consider a square matrix \[A = \left[ {\mathop a\nolimits_{ij} } \right]\]
Where $i$: row number and $j$: column number.
Step 2: Condition for skew symmetric matrix:
$A' = - A$
Here,$A'$is transpose of matrix A
i.e. \[\mathop a\nolimits_{ij} = - \mathop a\nolimits_{ji} \]
Step 3: Now, if we put $i = j$,
We have, \[\mathop a\nolimits_{ii} = - \mathop a\nolimits_{ii} \]
\[
  \therefore 2\mathop a\nolimits_{ii} = 0 \\
   \Rightarrow \mathop a\nolimits_{ii} = 0 \\
 \] for all $i's$.
Step 4: diagonal elements of a square matrix
In the square matrix \[A = \left[ {\mathop a\nolimits_{ij} } \right]\]
$A = \left( {\begin{array}{*{20}{c}}
  {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\
  {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\
  {{a_{31}}}&{{a_{32}}}&{{a_{33}}}
\end{array}} \right)$
Elements ${a_{11}},{a_{22}},{a_{33}}$ are diagonal elements.
${a_{ii}} = 0$
$ \Rightarrow {a_{11}} = {a_{22}} = {a_{33}} = 0$

All the diagonal elements of the skew symmetric matrix are zero. Thus, the correct option is (B).

Note: Another way to understand the solution.
We have a theorem: Any square matrix A with real number entries, $A - A'$is a skew symmetric matrix.
Example question: The skew symmetric matrix of matrix $B = \left[ {\begin{array}{*{20}{c}}
  2&{ - 2}&{ - 4} \\
  { - 1}&3&4 \\
  1&{ - 2}&{ - 3}
\end{array}} \right]$.