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# $(i)$ Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix.

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• A square matrix A=[a$_{ij}$] is said to be symmetric if A'=A that is $[a_{ij}]=[a_{ji}]$ for all possible value of i and j.
$(i)Given A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$
$a_{21}=a_{12}=-1$
$a_{31}=a_{13}=5$
$a_{32}=a_{23}=1$
$a_{11}=a_{22}=a_{33}$ are 1 2 3 respectively.
Hence $a_{ij}=a_{ji}$
Therefore A is a symmetric matrix.
$A '= \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}=A$
Thus A is a symmetric matrix.