Question

Let A be a square matrix. Which of the following is/are not skew-symmetric matrix/matrices?
(a)$A - {A^T}$ (b)${A^T} - A$
(c)$A{A^T} - {A^T}A$ (d)$A + {A^T}$, when$A$is skew-symmetric

Answer 1

Hint: For a matrix $A$ to be skew-symmetric ${A^T} = - A$ must be true.

Let $A$ be any square matrix A = \left( {\begin{array}{*{20}{c}} 1&2 \\\ 3&4 \end{array}} \right)
The condition for a matrix $A$ to be skew-symmetric is
If ${A^T} = - A$, then $A$ is skew-symmetric. …(1)
Let us consider all the given options one by one and find out if they are skew-symmetric or not.
Option (a) is $A - {A^T}$. For this to be skew-symmetric, from (1) the condition is
${\left( {A - {A^T}} \right)^T} = - \left( {A - {A^T}} \right)$ …(2)
Let us find ${A^T}$. The transpose of a matrix is written by interchanging the rows and columns into columns and rows.
{A^T} = \left( {\begin{array}{*{20}{c}} 1&3 \\\ 2&4 \end{array}} \right)
A - {A^T} = \left( {\begin{array}{*{20}{c}} 1&2 \\\ 3&4 \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 1&3 \\\ 2&4 \end{array}} \right) \\\ A - {A^T} = \left( {\begin{array}{*{20}{c}} 0&{ - 1} \\\ 1&0 \end{array}} \right) \\\
{\left( {A - {A^T}} \right)^T} = \left( {\begin{array}{*{20}{c}} 0&1 \\\ { - 1}&0 \end{array}} \right) …(3)
- \left( {A - {A^T}} \right) = - \left( {\begin{array}{*{20}{c}} 0&{ - 1} \\\ 1&0 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&1 \\\ { - 1}&0 \end{array}} \right) …(4)
From (2), (3), and (4), we find that ${\left( {A - {A^T}} \right)^T} = - \left( {A - {A^T}} \right)$
holds true. So, $A - {A^T}$ is skew-symmetric.
Option (b) is ${A^T} - A$. For this to be skew-symmetric, from (1) the condition is
${\left( {{A^T} - A} \right)^T} = - \left( {{A^T} - A} \right)$ …(5)
{A^T} - A = \left( {\begin{array}{*{20}{c}} 1&3 \\\ 2&4 \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 1&2 \\\ 3&4 \end{array}} \right) \\\ {A^T} - A = \left( {\begin{array}{*{20}{c}} 0&1 \\\ { - 1}&0 \end{array}} \right) \\\
{\left( {{A^T} - A} \right)^T} = \left( {\begin{array}{*{20}{c}} 0&{ - 1} \\\ 1&0 \end{array}} \right) …(6)
- \left( {{A^T} - A} \right) = - \left( {\begin{array}{*{20}{c}} 0&1 \\\ { - 1}&0 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&{ - 1} \\\ 1&0 \end{array}} \right) …(7)
From (5), (6), and (7), we find that ${\left( {{A^T} - A} \right)^T} = - \left( {{A^T} - A} \right)$
holds true. So, ${A^T} - A$ is skew-symmetric.
Option (c) is $A{A^T} - {A^T}A$. For this to be skew-symmetric, from (1) the condition is
${\left( {A{A^T} - {A^T}A} \right)^T} = - \left( {A{A^T} - {A^T}A} \right)$ …(8)
A{A^T} = \left( {\begin{array}{*{20}{c}} 1&2 \\\ 3&4 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&3 \\\ 2&4 \end{array}} \right) \\\ A{A^T} = \left( {\begin{array}{*{20}{c}} {1\left( 1 \right) + 2\left( 2 \right)}&{1\left( 3 \right) + 2\left( 4 \right)} \\\ {3\left( 1 \right) + 4\left( 2 \right)}&{3\left( 3 \right) + 4\left( 4 \right)} \end{array}} \right) \\\ A{A^T} = \left( {\begin{array}{*{20}{c}} 5&{11} \\\ {11}&{25} \end{array}} \right) \\\
{A^T}A = \left( {\begin{array}{*{20}{c}} 1&3 \\\ 2&4 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&2 \\\ 3&4 \end{array}} \right) \\\ {A^T}A = \left( {\begin{array}{*{20}{c}} {1\left( 1 \right) + 3\left( 3 \right)}&{1\left( 2 \right) + 3\left( 4 \right)} \\\ {2\left( 1 \right) + 4\left( 3 \right)}&{2\left( 2 \right) + 4\left( 4 \right)} \end{array}} \right) \\\ {A^T}A = \left( {\begin{array}{*{20}{c}} {10}&{14} \\\ {14}&{20} \end{array}} \right) \\\
A{A^T} - {A^T}A = \left( {\begin{array}{*{20}{c}} 5&{11} \\\ {11}&{25} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} {10}&{14} \\\ {14}&{20} \end{array}} \right) \\\ A{A^T} - {A^T}A = \left( {\begin{array}{*{20}{c}} { - 5}&{ - 3} \\\ { - 3}&5 \end{array}} \right) \\\
{\left( {A{A^T} - {A^T}A} \right)^T} = \left( {\begin{array}{*{20}{c}} { - 5}&{ - 3} \\\ { - 3}&5 \end{array}} \right) …(9)
- \left( {A{A^T} - {A^T}A} \right) = - \left( {\begin{array}{*{20}{c}} { - 5}&{ - 3} \\\ { - 3}&5 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 5&3 \\\ 3&{ - 5} \end{array}} \right) …(10)
From (8), (9), and (10), we find that ${\left( {A{A^T} - {A^T}A} \right)^T} \ne - \left( {A{A^T} - {A^T}A} \right)$. So, $A{A^T} - {A^T}A$ is not skew-symmetric.
Option (d) is $A + {A^T}$, when $A$ is skew-symmetric. It is given that $A$ is skew-symmetric. So, from (1),
${A^T} = - A$ …(11)
For $A + {A^T}$ to be skew-symmetric, we need to prove that ${\left( {A + {A^T}} \right)^T} = \- \left( {A + {A^T}} \right)$
Using (11) here, we get
LHS = {(A + {A^T})^T} = {(A + {( - A)^T})^T} = \left( {\begin{array}{*{20}{c}} 0&0 \\\ 0&0 \end{array}} \right) \\\ RHS = - (A + {A^T}) = - (A + ( - A)) = \left( {\begin{array}{*{20}{c}} 0&0 \\\ 0&0 \end{array}} \right) \\\
LHS=RHS
So, ${\left( {A + {A^T}} \right)^T} = - \left( {A + {A^T}} \right)$ holds true. Hence, $A + {A^T}$ is skew-symmetric, when $A$ is skew-symmetric.
Hence, the only option (c) is not skew-symmetric.

Note: This problem can be alternatively solved without using the sample matrix and just by
using the formula ${\left( {A + B} \right)^T} = {A^T} + {B^T}$ for each of the statements given
to prove the condition for a skew-symmetric matrix. For example, for this $A - {A^T}$ option,
the condition is ${\left( {A - {A^T}} \right)^T} = - \left( {A - {A^T}} \right)$. $LHS = {\left( {A - {A^T}} \right)^T} = {A^T} - {\left( {{A^T}} \right)^T} = {A^T} - A = - \left( {A - {A^T}} \right) = RHS$. Similarly, this can be done for all options to prove it.
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