ISRO | ISRO CS 2017 | Question 17

If C is a skew-symmetric matrix of order n and X is n x 1 column matrix, then X^TCX is a

(A) scalar matrix
(B) null matrix
(C) unit matrix
(D) matrix will all elements 1


Answer: (B)

Explanation: A is skew symmetric of order n X n,order of X = n x 1 and order of X’ = 1 x n, then order of

X'AX = ((1 X n)(n X n))(n X 1) 
= (1 X n)(n X 1) = 1

Let X'AX = Y
(X'AX)' = Y'

X'A'X" = Y' (since (AB)' = B'A')

As A is a skew-symmetric matrix, A' = -A 

X'(-A)X" = Y'
-(X'AX) = Y
-Y = Y
 Y = 0

So, option (B) is correct.

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