Question

Let A, B, C, D be (not necessarily square) real matrices such that, \[{{A}^{T}}=BCD;{{B}^{T}}=CDA\]; \[{{C}^{T}}=DAB\] and \[{{D}^{T}}=ABC\]. For the matrix S = ABCD, consider the two statements,
(i) \[{{S}^{3}}=S\]
(ii) \[{{S}^{2}}={{S}^{4}}\]
(A) (ii) is true but not (i)
(B) (i) is true but not (ii)
(C) both (i) and (ii) are true
(D) both (i) and (ii) are false

Answer 1

Hint: Apply the formula: - \[{{\left( MN \right)}^{T}}={{N}^{T}}.{{M}^{T}}\] and multiply \[{{D}^{T}},{{C}^{T}},{{B}^{T}},{{A}^{T}}\] to get \[{{\left( ABCD \right)}^{T}}\] in the L.H.S. Now, substitute ABCD = S as given in the question to check whether the conditions given in statement (i) and (ii) are correct or not.

Complete step by step answer:
We have been provided with the relations: -
\[\begin{align}
  & {{A}^{T}}=BCD \\
 & {{B}^{T}}=CDA \\
\end{align}\]
\[{{C}^{T}}=DAB\]
\[{{D}^{T}}=ABC\]
Here, \[{{A}^{T}},{{B}^{T}},{{C}^{T}},{{D}^{T}}\] represents the transpose of A, B, C, D respectively. Now, multiplying \[{{D}^{T}},{{C}^{T}},{{B}^{T}},{{A}^{T}}\] in this order, we get,
\[{{D}^{T}}{{C}^{T}}{{B}^{T}}{{A}^{T}}=ABC.DAB.CDA.BCD\]
This can be simplified as: -
\[\Rightarrow {{D}^{T}}{{C}^{T}}{{B}^{T}}{{A}^{T}}=\left( ABCD \right).\left( ABCD \right).\left( ABCD \right)\]
\[\Rightarrow {{D}^{T}}{{C}^{T}}{{B}^{T}}{{A}^{T}}=S.S.S\], since ABCD = S
\[\Rightarrow {{D}^{T}}{{C}^{T}}{{B}^{T}}{{A}^{T}}={{S}^{3}}\] - (1)
Now, we know that, \[{{\left( MN \right)}^{T}}={{N}^{T}}{{M}^{T}}\],
\[\begin{align}
  & \Rightarrow {{D}^{T}}{{C}^{T}}{{B}^{T}}{{A}^{T}}={{\left( ABCD \right)}^{T}} \\
 & \Rightarrow {{D}^{T}}{{C}^{T}}{{B}^{T}}{{A}^{T}}={{S}^{T}} \\
\end{align}\]
Substituting this value in equation (1), we have,
\[\Rightarrow {{S}^{T}}={{S}^{3}}\]
We can see that in the question we haven’t been informed about \[{{S}^{T}}\], whether it is equal to S or not. Therefore, we can conclude that statement (i), \[{{S}^{3}}=S\] is false.
Now, considering statement (ii), we have,
\[\Rightarrow {{S}^{4}}={{S}^{3}}.S\]
\[\Rightarrow {{S}^{4}}={{S}^{T}}.S\], since \[{{S}^{T}}={{S}^{3}}\] is shown above.
Now, \[\Rightarrow {{S}^{T}}\ne S\Rightarrow {{S}^{T}}.S\ne {{S}^{2}}\]. So,
\[{{S}^{4}}\ne {{S}^{2}}\], therefore statement (ii) is also false.

So, the correct answer is “Option D”.

Note: One may note that we have found \[{{S}^{T}}={{S}^{3}}\] and not \[S={{S}^{3}}\], so statement (i) was considered false. If we were provided with the information that S is a symmetric matrix then we would have used the relation \[{{S}^{T}}=S\] and statement (i) would have turned out to be true. Once statement (i) gets true statement (ii) would also get true. So, remember that we need information about S whether it is symmetric or not.