JEE Advanced | 2015MatricesHard
Question
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 ?
Options
A.Y3Z4 − Z4Y3
B.X44 + Y44
C.X4Z3 − Z3X4
D.X23 + Y23
Solution
xT = − x, yT = − y, zT = z
(A) Let P = y3z4 − z4y3
PT = (y3z4)T − (z4y3)T
= − z4y3 + y3 z4 = P ⇒ symmetric
(B) Let P = x44 + y44
PT = (X44)T + (y44)T = P ⇒ symmetric
(C) Let P = x4z3 – z3x4
PT = (z3)T(x4)T − (x4)T(z3)T
= z3x4 − x4z3 = − P ⇒ skew symmetric
(D) Let P = x23 + y23
PT = −x23 − y23 = − P ⇒ skew symmetric
(A) Let P = y3z4 − z4y3
PT = (y3z4)T − (z4y3)T
= − z4y3 + y3 z4 = P ⇒ symmetric
(B) Let P = x44 + y44
PT = (X44)T + (y44)T = P ⇒ symmetric
(C) Let P = x4z3 – z3x4
PT = (z3)T(x4)T − (x4)T(z3)T
= z3x4 − x4z3 = − P ⇒ skew symmetric
(D) Let P = x23 + y23
PT = −x23 − y23 = − P ⇒ skew symmetric
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