JEE MAIN - Mathematics (2023 - 25th January Evening Shift - No. 6)
Let A, B, C be 3 $$\times$$ 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements
(S1) A$$^{13}$$ B$$^{26}$$ $$-$$ B$$^{26}$$ A$$^{13}$$ is symmetric
(S2) A$$^{26}$$ C$$^{13}$$ $$-$$ C$$^{13}$$ A$$^{26}$$ is symmetric
Then,
Only S2 is true
Only S1 is true
Both S1 and S2 are false
Both S1 and S2 are true
Explanation
$A^{T}=A, B^{T}=-B, C^{T}=-C$
$$ \begin{aligned} P & =A^{13} B^{26}-B^{26} A^{13} \\\\ P^{T} & =\left(A^{13} B^{26}-B^{26} A^{13}\right)^{T}=\left(A^{13} B^{26}\right)^{T}-\left(B^{26} A^{B}\right)^{T} \\\\ & =\left(B^{26}\right)^{T}\left(A^{13}\right)^{T}-\left(A^{13}\right)^{T}\left(B^{26}\right)^{T} \\\\ & =\left(B^{T}\right)^{26}\left(A^{T}\right)^{13}-\left(A^{T}\right)^{13}\left(A^{T}\right)^{26} \\\\ & =B^{26} A^{13}-A^{13} B^{26}=-\left(A^{13} B^{26}-B^{26} A^{13}\right)=-P \end{aligned} $$
$P$ is skew-symmetric matrix $\Rightarrow S_{1}$ is false
$Q=A^{26} C^{13}-C^{13} A^{26}=Q^{T}=\left(A^{26} C^{13}-C^{13} A^{26}\right)^{T}$
$Q=\left(A^{26} C^{13}\right)^{T}-\left(C^{13} A^{26}\right)^{T}=\left(C^{13}\right)^{T}\left(A^{26}\right)^{T}-\left(A^{26}\right)^{T}\left(C^{13}\right)^{T}$
$=\left(C^{T}\right)^{13}\left(A^{T}\right)^{26}-\left(A^{T}\right)^{26}\left(C^{T}\right)^{13}=-C^{13} A^{26}+A^{26} C^{13}$
$=A^{26} C^{13}+C^{13} A^{26}$
$\Rightarrow Q^{T}=Q \Rightarrow Q$ is symmetric matrix $\Rightarrow S_{2}$ is true.
$$ \begin{aligned} P & =A^{13} B^{26}-B^{26} A^{13} \\\\ P^{T} & =\left(A^{13} B^{26}-B^{26} A^{13}\right)^{T}=\left(A^{13} B^{26}\right)^{T}-\left(B^{26} A^{B}\right)^{T} \\\\ & =\left(B^{26}\right)^{T}\left(A^{13}\right)^{T}-\left(A^{13}\right)^{T}\left(B^{26}\right)^{T} \\\\ & =\left(B^{T}\right)^{26}\left(A^{T}\right)^{13}-\left(A^{T}\right)^{13}\left(A^{T}\right)^{26} \\\\ & =B^{26} A^{13}-A^{13} B^{26}=-\left(A^{13} B^{26}-B^{26} A^{13}\right)=-P \end{aligned} $$
$P$ is skew-symmetric matrix $\Rightarrow S_{1}$ is false
$Q=A^{26} C^{13}-C^{13} A^{26}=Q^{T}=\left(A^{26} C^{13}-C^{13} A^{26}\right)^{T}$
$Q=\left(A^{26} C^{13}\right)^{T}-\left(C^{13} A^{26}\right)^{T}=\left(C^{13}\right)^{T}\left(A^{26}\right)^{T}-\left(A^{26}\right)^{T}\left(C^{13}\right)^{T}$
$=\left(C^{T}\right)^{13}\left(A^{T}\right)^{26}-\left(A^{T}\right)^{26}\left(C^{T}\right)^{13}=-C^{13} A^{26}+A^{26} C^{13}$
$=A^{26} C^{13}+C^{13} A^{26}$
$\Rightarrow Q^{T}=Q \Rightarrow Q$ is symmetric matrix $\Rightarrow S_{2}$ is true.
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