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  • Let A,B,C be 3 × 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements <img alt="(S1) \mathrm{A}^{13} \mathrm{~B}^{26}-\mathrm{B}^{26} \mathrm{

Let A,B,C be 3 × 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements

(S1) \mathrm{A}^{13} \mathrm{~B}^{26}-\mathrm{B}^{26} \mathrm{~A}^{13}  is symmetric

(\mathrm{S} 2) \mathrm{A}^{26} \mathrm{C}^{13}-\mathrm{C}^{13} \mathrm{~A}^{26} is symmetric

Then,
 

Option: 1

Only S2 is true


Option: 2

Both S1 and S2 are false

 


Option: 3

Only S1 is true


Option: 4

Both S1 and S2 are true


Answers (1)

best_answer

\begin{aligned} & \mathrm{A}^{\mathrm{T}}=\mathrm{A}, \quad \mathrm{B}^{\mathrm{T}}=-\mathrm{B}, \quad \mathrm{C}^{\mathrm{T}}=-\mathrm{C} \\ & \left(\mathrm{S}_1\right): \quad\left(\mathrm{A}^{13} \mathrm{~B}^{26}-\mathrm{B}^{26} \mathrm{~A}^{13}\right)^{\mathrm{T}} \\ & =\left(A^{13} B^{26}\right)^T-\left(\mathrm{B}^{26} \mathrm{~A}^{13}\right)^{\mathrm{T}} \\ & =\left(\mathrm{B}^{\mathrm{T}}\right)^{26}\left(\mathrm{~A}^{\mathrm{T}}\right)^{13}-\left(\mathrm{A}^{\mathrm{T}}\right)^{13}\left(\mathrm{~B}^{\mathrm{T}}\right)^{26} \\ & =(-\mathrm{B})^{26}(\mathrm{~A})^{13}-(\mathrm{A})^{13}(-\mathrm{B})^{26} \\ & =\mathrm{B}^{26} \mathrm{~A}^{13}-\mathrm{A}^{13} \mathrm{~B}^{26} \\ & =-\left(A^{13} B^{26}-B^{26} A^{13}\right) \\ & \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \left\(\mathrm{S}_1 \rightarrow \text { false }\right) \\ & \end{aligned}

\begin{aligned} \left(\mathrm{S}_2\right): & \left(\mathrm{A}^{26} \mathrm{C}^{13}-\mathrm{C}^{13} \mathrm{~A}^{26}\right)^{\mathrm{T}} \\ & =\left(\mathrm{A}^{26} \mathrm{C}^{13}\right)^{\mathrm{T}}-\left(\mathrm{C}^{13} \mathrm{~A}^{26}\right)^{\mathrm{T}} \\ & =\left(\mathrm{C}^{\mathrm{T}}\right)^{13}\left(\mathrm{~A}^{\mathrm{T}}\right)^{26}-\left(\mathrm{A}^{\mathrm{T}}\right)^{26}\left(\mathrm{C}^{\mathrm{T}}\right)^{13} \end{aligned}

\begin{aligned} &\begin{aligned} & =-C^{13} A^{26}-A^{26}(-C)^{13} \\ & =A^{26} C^{13}-C^{13} A^{26} \end{aligned}\\ &\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \left(\mathrm{S}_2 \rightarrow \text { True }\right) \end{aligned}

Posted by

Pankaj Sanodiya

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