Matrices Question 6
Question 6 - 25 January - Shift 2
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,
(1) Only S2 is true
(2) Only S1 is true
(3) Both $S 1$ and $S 2$ are false
(4) Both S1 and S2 are true
Show Answer
Answer: (1)
Solution:
Formula: Properties of Transpose of matrix, Properties Of Positive Integral Powers Of A Square Matrix, Types of matrix
Given, $A^{T}=A, B^{T}=-B, C^{T}=-C$
Let $M=A^{13} B^{26}-B^{26} A^{13}$
Then, $M^{T}=(A^{13} B^{26}-B^{26} A^{13})^{T}$
$=(A^{13} B^{26})^{T}-(B^{26} A^{13})^{T}$
$=(B^{T})^{26}(A^{T})^{13}-(A^{T})^{13}(B^{T})^{26}$
$=B^{26} A^{13}-A^{13} B^{26}=-M$
Hence, $M$ is skew symmetric
Let, $N=A^{26} C^{13}-C^{13} A^{26}$
then, $N^{T}=(A^{26} C^{13})^{T}-(C^{13} A^{26})^{T}$
$=-(C)^{13}(A)^{26}+A^{26} C^{13}=N$
Hence, $N$ is symmetric.
$\therefore$ Only $S 2$ is true.
