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Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMatrices2 Marks
Question
For the matrix $A=\left[\begin{array}{ll} {1} & {5} \\ {6} & {7} \end{array}\right]$, verify that (A + A′) is a symmetric matrix.
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Answer

Here,
$A=\left[\begin{array}{ll} {1} & {5} \\ {6} & {7} \end{array}\right] \Rightarrow A^{\prime}=\left[\begin{array}{ll} {1} & {6} \\ {5} & {7} \end{array}\right]$
On adding them we get,
$A+A^{\prime}=\left[\begin{array}{ll} {1} & {5} \\ {6} & {7} \end{array}\right]+\left[\begin{array}{ll} {1} & {6} \\ {5} & {7} \end{array}\right]$
$\Rightarrow A+A^{\prime}=\left[\begin{array}{cc} {1+1} & {5+6} \\ {6+5} & {7+7} \end{array}\right]$
$\Rightarrow \mathrm{A}+\mathrm{A}^{\prime}=\left[\begin{array}{ll} {2} & {11} \\ {11} & {14} \end{array}\right] $ ...(1)
Now, $(\mathrm{A}+\mathrm{A}^{\prime})^\prime=A^\prime+A=$ $\left(\mathrm{A}+\mathrm{A}^{\prime}\right)=\left[\begin{array}{ll} {2} & {11} \\ {11} & {14} \end{array}\right] $ ...(2)
So, from equation (1) & (2) we get,
(A + A') = (A + A')', hence we can say that (A + A') is a Symmetric matrix.

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