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  • If matrix A is skew-symmetric then,Option: 1 <img alt="a_{ij}=-a_{ji}" src="https://entrancecorner.oncodecogs.com/gif.latex?a_%7Bij%7D%3D-a_%7Bji%7D"

If matrix A is skew-symmetric then,

Option: 1

a_{ij}=-a_{ji} if j\neq i and a_{ij}=a_{ji} if j= i


Option: 2

a_{ij}\neq-a_{ji} if j\neq i and a_{ij}=a_{ji} if j= i


Option: 3

a_{ij}=-a_{ji} if j\neq i and a_{ij}=0 if j= i


Option: 4

a_{ij}\neq-a_{ji} if j\neq i and a_{ij}=0 if j= i


Answers (1)

best_answer

 

 

symmetric and skew symmetric matrix -

symmetric matrix: A square matrix \\\mathrm{A=\left [ a_{ij} \right ]_{n\times n}} is said to be symmetric if A' = A,

    \\\mathrm{ i.e., a_{ij} = a_{ji}\; \forall\; i, j}

    \\\mathrm{A=\begin{bmatrix} a & h & g\\ h& b & f\\ g& f & c \end{bmatrix}\; then \; A'=\begin{bmatrix} a & h & g\\ h & b & f\\ g & f & c \end{bmatrix}}

 

Skew-symmetric matrix:  A square matrix  \\\mathrm{A=\left [ a_{ij} \right ]_{m\times n}} is said to be skew-symmetric if A’ = -A 

\\\mathrm{i.e. A' = -A , i.e., a_{ij} =- a_{ji}\; \forall \;i, j } \\\mathrm{Now \;if \;we\; put\; i\; =\; j,\; we\; have} \\\mathrm{ a_{ii} = -a_{ii},} \\\mathrm{ \therefore 2a_{ii}=0 \Rightarrow a_{ii}=0\; \forall \; i's}

That means all the diagonal element of the skew-symmetric matrix is 0 but not the other way around.

\\\mathrm{e.g.\;\; A=\begin{bmatrix} 0 & h & g\\ -h & 0 & f\\ -g & -f & 0 \end{bmatrix},\; then\; A' = \begin{bmatrix} 0 & -h & -g\\ h & 0 & -f\\ g & f & 0 \end{bmatrix} = -A}

-

 

 

In a skew-symmetric matrix diagonal element is zero

And a_{ij}=a_{ji}

Posted by

Rakesh

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