Question

Let $X$ and $Y$ be two arbitrary,$3\times 3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3\times 3$ , non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?
(a) ${{Y}^{3}}{{Z}^{4}}-{{Z}^{4}}{{Y}^{3}}$
(b) ${{X}^{44}}-{{Y}^{44}}$
(c) ${{X}^{4}}{{Z}^{3}}-{{Z}^{3}}{{X}^{4}}$
(d) ${{X}^{23}}+{{Y}^{23}}$

Answer 1

Hint: To solve the above question, we have to use the concept of Transpose of matrix. The properties of
Transpose of matrix is ${{\left( A+B \right)}^{T}}={{A}^{T}}+{{B}^{T}}$ and ${{\left( AB \right)}^{T}}={{B}^{T}}{{A}^{T}}$ .We also has to know the concept of
Skew symmetric and symmetric matrix.

Complete step by step answer:
Now we are using the properties of transpose of matrix that is given in below,
${{\left( A+B \right)}^{T}}={{A}^{T}}+{{B}^{T}}$and ${{\left( AB \right)}^{T}}={{B}^{T}}{{A}^{T}}$
Now we consider the option C:
$\left( {{X}^{4}}{{Z}^{3}}-{{Z}^{3}}{{X}^{4}} \right)^T={{\left( {{X}^{4}}{{Z}^{3}} \right)}^{T}}-{{\left( {{Z}^{3}}{{X}^{4}} \right)}^{T}}$
$={{\left( {{Z}^{3}} \right)}^{T}}{{\left( {{X}^{4}} \right)}^{T}}-{{\left( {{X}^{4}} \right)}^{T}}{{\left( {{Z}^{3}} \right)}^{T}}={{\left( {{Z}^{T}} \right)}^{3}}{{\left( {{X}^{T}} \right)}^{4}}-{{\left( {{X}^{T}} \right)}^{4}}{{\left( {{Z}^{T}} \right)}^{3}}={{Z}^{3}}{{X}^{4}}-{{X}^{4}}{{Z}^{3}}$
If we simplify it we will get it as
$=-\left( {{X}^{4}}{{Z}^{3}}-{{Z}^{3}}{{X}^{4}} \right)$
So, we can see that it is a skew symmetric matrix.
Now we consider the option A:
${{\left( {{Y}^{3}}{{Z}^{4}}-{{Z}^{4}}{{Y}^{3}} \right)}^{T}}={{\left( {{Y}^{3}}{{Z}^{4}} \right)}^{T}}-{{\left( {{Z}^{4}}{{Y}^{3}} \right)}^{T}}$
$={{\left( {{Z}^{4}} \right)}^{T}}{{\left( {{Y}^{3}} \right)}^{T}}-{{\left( {{Y}^{3}} \right)}^{T}}{{\left( {{Z}^{4}} \right)}^{T}}={{\left( {{Z}^{T}} \right)}^{4}}{{\left( {{Y}^{T}} \right)}^{3}}-{{\left( {{Y}^{T}} \right)}^{3}}{{\left( {{Z}^{T}} \right)}^{4}}={{Y}^{3}}{{Z}^{4}}-{{Z}^{4}}{{Y}^{3}}$
So, we can see that it is a symmetric matrix.
Now we consider the option B:
For this case we can see that
$\left( {{X}^{44}}+{{Y}^{44}} \right)^T={{\left( {{X}^{T}} \right)}^{44}}+{{\left( {{Y}^{T}} \right)}^{44}}=\left( {{X}^{44}}+{{Y}^{44}} \right)$
So, we can see that it is a symmetric matrix.
Now we consider the option D:
$\left( {{X}^{23}}+{{Y}^{23}} \right)^T={{\left( {{X}^{T}} \right)}^{23}}+{{\left( {{Y}^{T}} \right)}^{23}}=-\left( {{X}^{23}}+{{Y}^{23}} \right)$
So, we can see that it is a skew symmetric matrix.

Hence we can see that option (d) and (c) is a skew symmetric matrix and option (b) and (a) is a symmetric matrix.

Note: Here student must take care of the concept of Transpose of matrix and also concept of symmetric
And skew a symmetric matrix. Students sometimes make mistakes between symmetric and skew symmetric matrices. They think they are same but they are different because the condition of a matrix like $A$ to be symmetric if $A={{A}^{T}}$ and the condition of a matrix like $A$ to be skew symmetric if $A={{A}^{T}}$.