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Home  >>  CBSE XII  >>  Math  >>  Matrices
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$ (ii)$ Show that the matrix $ A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}$ is a skew symmetric matrix. $

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  • A square matrix A=[a$_{ij}$] is said to be skew symmetric if A'=-A that is $[a_{ij}]= -[a_{ji}]$ for all possible value of i and j.
  • All diagonal element of a skew symmetric matrix are zero.
(ii)A square matrix $A=[a_{ij}]$ is said to be skew symmetric matrix if A'=-A that is $a_{ij}=-a_{ij}$ for all possible values of i and j.Now if we put i=j we have $a_{ii}=-a_{ii}\Rightarrow 2a_{ii}=0$
$a_{ii}=0.$
For a skew symmetric matrix $a_{ji}=-a_{ij}.$
Given $A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}$
$a_{21}=-1\Rightarrow a_{12}=1$
$-a_{12}=-1\Rightarrow a_{21}=-a_{12}$
$a_{31}=1\Rightarrow a_{12}=-1$
$-a_{13}=1\Rightarrow a_{31}=-a_{13}$
$a_{32}=-1\Rightarrow a_{23}=1$
$-a_{23}=-1\Rightarrow a_{32}=-a_{23}$
$a_{11}=0,a_{22}=0,a_{33}=0$
$ A' = \begin{bmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{bmatrix} $
$ A' =(-1) \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix} $
All diagonal element of the above matrix are zero. Hence it is a skew symmetric matrix
$\Rightarrow A'=(-1)A.$
Thus A is a skew symmetric matrix.
answered Mar 14, 2013 by sharmaaparna1
 

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