Q29 / 60 · JEE Advanced 2015 (Paper 1) · Mathematics

Let X and Y be two arbitrary, 3 ×3 , non - zero, skew - symmetric matrices and Z be an arbitrary 3 ×3 , non - zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?

  • A
    Y3 Z4-Z4 Y3
  • B
    X44+Y44
  • Correct answer
    C
    X4Z3-Z3X4
  • Correct answer
    D
    X23+Y23

Solution

XT=-X, YT=-Y and ZT=Z
P=Y3Z4-Z4Y3
PT=Z4TY3T-Y3TZ4T
= ZT4YT3-YT3ZT4
= Z4- Y3--Y3Z4
= Y3Z4 – Z4Y3 = P(symmetric) 
Q=X44+Y44
QT=XT44+YT44=X44+Y44= Q(symmetric)
RT=Z3TX4T-X4TZ3T
= ZT3XT4-XT4 ZT3
= Z3-X4--X4Z3
= Z3X4-X4Z3=-R (Skew Symmetric)
S=X23+Y23
ST=XT23+YT23=-X23+-Y23
=-X23-Y23=-S (Skew symmetric)

Hence, the correct answer is options: 'C', 'D'.

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