If C is a skew-symmetric matrix of order n and X is n x 1 column matrix, then XTCX is a
(A) scalar matrix
(B) null matrix
(C) unit matrix
(D) matrix will all elements 1
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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