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Prove that a − at is a Skew-symmetric Matrix. - Mathematics

Sum

If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − AT is a skew-symmetric matrix.

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Solution

\[Given: \hspace{0.167em} A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix} \]
\[ A^T = \begin{bmatrix}2 & 4 \\ 3 & 5\end{bmatrix}\]
\[Now, \]
\[\left( A - A^T \right) = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix} - \begin{bmatrix}2 & 4 \\ 3 & 5\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right) = \begin{bmatrix}2 - 2 & 3 - 4 \\ 4 - 3 & 5 - 5\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right) = \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix} . . . \left( 1 \right)\]
\[ \left( A - A^T \right)^T = \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix}^T \]
\[ \Rightarrow \left( A - A^T \right)^T = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right)^T = - \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right) = - \left( A - A^T \right)^T \left[ \text{Using eq} . \left( 1 \right) \right]\]
\[Thus, \left( A - A^T \right) \text{is a skew - symmetric matrix} .\]

Concept: Types of Matrices
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 5 Algebra of Matrices
Exercise 5.5 | Q 1 | Page 60
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