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6 votes
6 votes

If $C$ is a skew-symmetric matrix of order $n$ and $X$ is $n\times 1$ column matrix, then $X{^T} CX$ is a

  1. scalar matrix
  2. null matrix
  3. unit matrix
  4. matrix will all elements $1$
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3 Answers

Best answer
14 votes
14 votes
Let $C = \begin{bmatrix} 0&B &C \\ -B&0 &D \\ -C&-D&0 \end{bmatrix}$, and $X = \begin{bmatrix} P\\ Q\\ R \end{bmatrix}$, then

$CX$=$\begin{bmatrix} BQ+CR\\ -BP+DR\\ -CP-DQ \end{bmatrix}$

$X^{T}CX=$$\begin{bmatrix} P & Q &R \end{bmatrix}\times \begin{bmatrix}BQ+CR\\-BP+DR\\-CP-DQ \end{bmatrix}$

$\rightarrow \begin{bmatrix} PBQ+PCR-PBQ+QDR-PCR-QDR \end{bmatrix} \rightarrow\begin{bmatrix}PBQ-PBQ+PCR-PCR+QDR-QDR \end{bmatrix}$

$\rightarrow\begin{bmatrix}0\end{bmatrix}$

Option B.
selected by
13 votes
13 votes
Let $K=X^{T}CX   $  [K will be of 1X1 as $X^{T}: $1*n C:n*n and X:n*1]

As K is 1*1 i.e single element so $K=K^{T}$

$ K^{T}=(X^{T}CX) ^{T} $

$   K      =(X)^{T}C^{T}(X^{T})^{T}  $

        $=X^{T}(-C)X$ [As C is skew Symmetric so $C^{T}=-C$ ]

         $=-X^{T}CX$

         $             =        -K$

$2K      =     0$

$K=0$

$X^{T}CX=0$

So B)Null matrix should be answer.
Answer:

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