in Linear Algebra recategorized by
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6 votes
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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 $1$
in Linear Algebra recategorized by
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4 Comments

Order : $((1XN.NXN).NX1)=1XN.NX1=1X1 \ unit \ matrix$
How skew symmetric property would come into picture. Please correct me !
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m also getting same ...how it could be null
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null matrix

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Ans: b

Clearly, XTCX is skew-symmetric so main diagonal elements must be zero and hence a,c,d can't be the answer.

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3 Answers

14 votes
14 votes
Best answer
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.

1 comment

Nice approach..
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8 votes
8 votes

pls go through the link.. I think option B is correct.

https://yutsumura.com/7-problems-on-skew-symmetric-matrices/

1 comment

Good problems/observations.
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Answer:

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