Question

Let A,B,C be 3×3 matrices such that A is symmetric and B and C are skew symmetric. Consider the statements:
(S1) A¹³B²⁶ − B²⁶A¹³ is symmetric.
(S2) A²⁶C¹³ − C¹³A²⁶ is symmetric.Then:

• A. Only $S 1$ is true
• B. Both S1 and S2 are false
• C. Both S1 and S2 are true
• D. Only S2 is true

Answer 1

The Correct Option is D

For statement S1:

\( A^{13}B^{26} - B^{26}A^{13} \) Given that \( A \) is symmetric and \( B, C \) are skew-symmetric, products involving an odd number of skew-symmetric matrices are skew-symmetric. Thus, \( B^2 \) and \( C^3 \) are skew-symmetric, making the whole expression skew-symmetric, and hence S1 is false.

For statement S2:

\( A^{26}C^{13} - C^{13}A^{26} \) Here, \( A^2 \) is symmetric and \( C^3C_1 \) (assuming \( C_1 = C \)) is also skew-symmetric. The product of a symmetric matrix with a skew-symmetric matrix, enclosed symmetrically, results in a symmetric matrix, so S2 is true.

Answer 2

The correct answer is (D) : Only S2 is true
Given, $A^T=A,B^T=-B,C^T=-C$
Let $M=A^{13}{B}^{26}-B^{26}{A}^{13}$
Then, $M^T={\left(A^{13}{B}^{26}-B^{26}{A}^{13}\right)}^T$
$={\left(A^{13}{B}^{26}\right)}^T-{\left(B^{26}{A}^{13}\right)}^T$
$={\left(B^T\right)}^{26}{\left(A^T\right)}^{13}-{\left(A^T\right)}^{13}{\left(B^T\right)}^{26}$
$=B^{26}{A}^{13}-A^{13}{B}^{26}=-M$
Hence, $M$ is skew symmetric
Let, $N=A^{26}{C}^{13}-C^{13}{A}^{26}$
then, $N^T={\left(A^{26}{C}^{13}\right)}^T-{\left(C^{13}{A}^{26}\right)}^T$
$=-(C{)}^{13}(A{)}^{26}+A^{26}{C}^{13}=N$
Hence, $N$ is symmetric.
$\therefore$ Only S2 is true.