Let A, B, C be 3 × 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,

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
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, [2023]

1 Only S1 is true

2 Both S1 and S2 are false

3 Both S1 and S2 are true

4 Only S2 is true

Ans.
(4)

$Since A is symmetric. ∴ A' = A$

$and B and C are skew symmetric.$

$∴ B' = - B and C' = - C$

$S 1 : (A^{13} B^{26} - B^{26} A^{13})' = (A^{13} B^{26})' - (B^{26} A^{13})'$

$= {(B')}^{26} {(A')}^{13} - {(A')}^{13} {(B')}^{26} (∵ (X' Y') = (Y' X'))$

$= {(- B)}^{26} {(A)}^{13} - {(A)}^{13} {(- B)}^{26}$

$= B^{26} A^{13} - A^{13} B^{26} = - (A^{13} B^{26} - B^{26} A^{13})$

Which is skew symmetric.

$∴ S 1 is false.$

$S 2 : (A^{26} C^{13} - C^{13} A^{26})' = (A^{26} C^{13})' - (C^{13} A^{26})'$

$= {(C')}^{13} {(A')}^{26} - {(A')}^{26} {(C')}^{13}$

$= {(- C')}^{13} A^{26} - A^{26} {(- C)}^{13} = - C^{13} A^{26} + A^{26} C^{13}$

Which is symmetric.

$∴ S 2 is true.$

Want Us to Email you About Special Offers & Updates?