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

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 $B = \left[ {\begin{array}{*{20}{c}} 2&{ - 2}&{ - 4} \\\ { - 1}&3&4 \\\ 1&{ - 2}&{ - 3} \end{array}} \right]$.