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Algebra of Matrices — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices3 Marks
Question
Define a symmetric matrix. Prove that for $\text{A}=\begin{bmatrix}2&4 \\5&6 \end{bmatrix},$ $A + A^T$ is a symmetric matrix where $A^T​​​​​​​$​​​​​​​ is the transpose of $A$.
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Answer

A squrae matrix A is called a syammetric matrix, if $A^T = A$.
Given: $\text{A}=\begin{bmatrix}2&4 \\5&6 \end{bmatrix}$
$\text{A}^{\text{T}}=\begin{bmatrix}2&5 \\4&6 \end{bmatrix}$
Now, $\text{A}+\text{A}^{\text{T}}=\begin{bmatrix}2&4 \\5&6 \end{bmatrix}+\begin{bmatrix}2&5 \\4&6 \end{bmatrix}$
$\Rightarrow\text{A}+\text{A}^{\text{T}}=\begin{bmatrix}4 & 9 \\9&12 \end{bmatrix}\ \dots(\text{i})$ $(\text{A}+\text{A}^{\text{T}})^{\text{T}}=\begin{bmatrix}4 & 9 \\9&12 \end{bmatrix}^{\text{T}}$ $=\begin{bmatrix}4 & 9 \\9&12 \end{bmatrix}$
$\Rightarrow\text{A}+\text{A}^{\text{T}} $ [From eq. (1)]
$\therefore\ (\text{A}+\text{A}^{\text{T}})^{\text{T}}=(\text{A}+\text{A}^{\text{T}})$
Thus, $(A + A^T)$ is a symmetric matrix.

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