If A is a skew symmetric matrix of order n and C is a column matrix of order n × 1, then CTAC is:

Question

If A is a skew symmetric matrix of order n and C is a column matrix of order n × 1, then CTAC is: (A) an identity matrix of order n (B) an identity matrix of order 1 (C) a zero matrix of order 1 (D) none of these

Tardigrade Admin · Accepted Answer

Given: A be a skew symmetric matrix of order n. i.e. AT = - (A)n × n C be the column matrix of order n × 1 then CTAC is of order 1 × 1. Let CTAC = α, Consider (CT AC)T = CT ATC = CT(-A) C = - CT AC = - α ⇒ α = - α ⇒ α = 0 Thus, CT AC is zero matrix of order 1.

Q. If A is a skew symmetric matrix of order n and C is a column matrix of order n × 1, then CTAC is:

an identity matrix of order n

an identity matrix of order 1

a zero matrix of order 1

none of these

Given : A be a skew symmetric matrix of order n. i.e. AT=−(A)n×n​ C be the column matrix of order n × 1 then CTAC is of order 1 × 1. Let CTAC=α, Consider (CTAC)T=CTATC=CT(−A)C=−CTAC=−α ⇒α=−α⇒α=0 Thus, CTAC is zero matrix of order 1.

Q. If A is a skew symmetric matrix of order n and C is a column matrix of order n $\times$ 1, then $C^{T} A C$ is: