Express the following matrices as the sum of a symmetric and a skew symmetric matrix :
$[\begin{matrix} 3 & 5 1 & - 1 \end{matrix}]$

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Solution

$\begin{matrix} L e t A = [\begin{matrix} 3 & 5 1 & - 1 \end{matrix}] A^{'} = [\begin{matrix} 3 & 1 5 & - 1 \end{matrix}] \frac{1}{2} (A + A^{'}) = \frac{1}{2} ([\begin{matrix} 3 & 5 1 & - 1 \end{matrix}] + [\begin{matrix} 3 & 1 5 & - 1 \end{matrix}]) = \frac{1}{2} [\begin{matrix} 6 & 6 6 & - 2 \end{matrix}] = [\begin{matrix} 3 & 3 3 & - 1 \end{matrix}] \frac{1}{2} (A - A^{'}) = \frac{1}{2} ([\begin{matrix} 3 & 5 1 & - 1 \end{matrix}] - [\begin{matrix} 3 & 1 5 & - 1 \end{matrix}]) = \frac{1}{2} [\begin{matrix} 0 & 4 - 4 & 0 \end{matrix}] = [\begin{matrix} 0 & 2 - 2 & 0 \end{matrix}] L e t P = \frac{1}{2} (A + A^{'}) = [\begin{matrix} 3 & 3 3 & - 1 \end{matrix}] P^{'} = [\begin{matrix} 3 & 3 3 & - 1 \end{matrix}] = P S i n c e, P^{'} = P P i s a s y m m e t r i c m a t r i x A g a i n Q = \frac{1}{2} (A - A^{'}) = [\begin{matrix} 0 & 2 - 2 & 0 \end{matrix}] Q^{'} = [\begin{matrix} 0 & - 2 2 & 0 \end{matrix}] = - [\begin{matrix} 0 & 2 - 2 & 0 \end{matrix}] = - Q S i n c e, Q^{'} = Q q i s s k e w s y m m e t r i c m a r i x . N o w, P + Q = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'}) = A \end{matrix}$
Hence, $A$ is a sum of symmetric & skew symmetruc matrix.

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Similar questions

Q. Express the following matrices as the sum of a symmetric and a skew symmetric matrix :

⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦

Q. Write

A = [\begin{matrix} 3 & 5 1 & - 1 \end{matrix}]

as the sum of a symmetric and a skew-symmetric matrix.

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

$[\begin{matrix} 1 & 5 - 1 & 2 \end{matrix}]$

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

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Q. Express

⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 1 2 & 1 & - 3 1 & 1 & 1 \end{matrix} ⎤ ⎥ ⎦

as a sum of symmetric and skew symmetric matrices.

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Express the following matrices as the sum of a symmetric and a skew symmetric matrix : [351−1]

Express the following matrices as the sum of a symmetric and a skew symmetric matrix :
$[\begin{matrix} 3 & 5 1 & - 1 \end{matrix}]$