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# For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$ , verify that $(i) (A+A')$ is a symmetric matrix.

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## 1 Answer

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Toolbox:
• 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.
Given:
$(i)A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$
$A' = \begin{bmatrix} 1 & 6 \\ 5 & 7 \end{bmatrix}$
$A+A' = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix} +\begin{bmatrix} 1 & 6 \\ 5 & 7 \end{bmatrix}$
$\Rightarrow \begin{bmatrix} 1+1 & 5+6 \\ 6+5 & 7+7 \end{bmatrix}$
$\Rightarrow \begin{bmatrix} 2 & 11 \\ 11 & 14 \end{bmatrix}$
$a_{12}=a_{21}=11$
If a 2x2 matrix [A] has its diagonal element to be equal then the transpose obtained will be equal to matrix [A]
Hence A+A' is a symmetric matrix.
answered Mar 14, 2013

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