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ISRO-DEC2017-17

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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 $1$

4 Answers

Best answer
16 16 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.
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16 16 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.
0 0 votes
X^T C X. This is a 1x1 matrix, which is essentially a scalar.

Since X^T C X is a scalar, its transpose is itself. So:

(X^T C X)^T = X^T C X

also, using the properties of transpose, (X^T C X)^T = X^T C^T (X^T)^T = X^T C^T X.

C is skew-symmetric: C^T = -C, so X^T C^T X = X^T (-C) X = -X^T C X.

Therefore, I have X^T C X = -X^T C X, which implies 2 X^T C X = 0, so X^T C X = 0.

Since it's a 1 x 1 matrix, zero means it's a null matrix.
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