### 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.