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Question: Let \(X\) and \(Y\) be two arbitrary,\(3\times 3\), non-zero, skew-symmetric matrices and \(Z\) be a...

Let XX and YY be two arbitrary,3×33\times 3, non-zero, skew-symmetric matrices and ZZ be an arbitrary 3×33\times 3 , non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?
(a) Y3Z4−Z4Y3{{Y}^{3}}{{Z}^{4}}-{{Z}^{4}}{{Y}^{3}}
(b) X44−Y44{{X}^{44}}-{{Y}^{44}}
(c) X4Z3−Z3X4{{X}^{4}}{{Z}^{3}}-{{Z}^{3}}{{X}^{4}}
(d) X23+Y23{{X}^{23}}+{{Y}^{23}}

Explanation

Solution

To solve the above question, we have to use the concept of Transpose of matrix. The properties of
Transpose of matrix is (A+B)T=AT+BT{{\left( A+B \right)}^{T}}={{A}^{T}}+{{B}^{T}} and (AB)T=BTAT{{\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,
(A+B)T=AT+BT{{\left( A+B \right)}^{T}}={{A}^{T}}+{{B}^{T}}and (AB)T=BTAT{{\left( AB \right)}^{T}}={{B}^{T}}{{A}^{T}}
Now we consider the option C:
(X4Z3−Z3X4)T=(X4Z3)T−(Z3X4)T\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}}
=(Z3)T(X4)T−(X4)T(Z3)T=(ZT)3(XT)4−(XT)4(ZT)3=Z3X4−X4Z3={{\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
=−(X4Z3−Z3X4)=-\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:
(Y3Z4−Z4Y3)T=(Y3Z4)T−(Z4Y3)T{{\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}}
=(Z4)T(Y3)T−(Y3)T(Z4)T=(ZT)4(YT)3−(YT)3(ZT)4=Y3Z4−Z4Y3={{\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
(X44+Y44)T=(XT)44+(YT)44=(X44+Y44)\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:
(X23+Y23)T=(XT)23+(YT)23=−(X23+Y23)\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 AA to be symmetric if A=ATA={{A}^{T}} and the condition of a matrix like AA to be skew symmetric if A=ATA={{A}^{T}}.