Question

If A is skew-symmetric of order n and C is a column matrix of order n × 1, then C^T AC is …

(a) an identity matrix of order n 

(b) an identity matrix of order 1

(c) a zero matrix of order I 

(d) an Identity matrix of order 2

Answer 1

(c) a zero matrix of order I

Given A is of order n × n 

C is of order n × 1

So, CT is of order 1 × n

Now C^TAC is of order 

Let it be equal to (x) say 

Taking transpose on either sides

(C^T, AC)^T(x)^T.

(i.e.) C^T(A^T)(C) = x 

C^T(-A)(C) = x 

⇒ C^TAC = -x 

⇒ x = -x ⇒ 2x = 0 ⇒ x = 0