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Write a Square Matrix Which is Both Symmetric as Well as Skew-symmetric.

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Question

Write a square matrix which is both symmetric as well as skew-symmetric.

Sum
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Solution

\[Let A = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} \] 

\[ A^T = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\] 

`"Since"   A^T = A,  A  is  a  symmmetric  matrix `

\[Now, \] 

\[ - A = - \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} \] 

\[ \Rightarrow - A = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\] 

`"Since"    A^T = - A,   A  is  a  skew - symmetric  matrix . `

Thus,` A= [[0  0  ],[0  0]]  `is an example of a matrix that is both symmetric and skew - symmetric. 

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Chapter 4: Algebra of Matrices - Exercise 5.6 [Page 63]

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R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12
Chapter 4 Algebra of Matrices
Exercise 5.6 | Q 31 | Page 63

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