Express the following matrix as the sum of a symmetric and a skew symmetric matrix-i - 3 5- 1 -1 -ii - 6 -2 2- -2 3 -1- 2 -1 3 -iii - 3 3 -1- -2 -2 1- -4 -5 2 -iv - 1 5- -1 2 -

(i) Let nbsp;A= nbsp; [ 3 nbsp; amp; nbsp; 5; nbsp; 1 amp; 1 ] nbsp; nbsp;⇒ A' = [ 3 nbsp; amp; nbsp; 1; nbsp; 5 amp; 1 ] nbsp ...

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Byju's Answer

Standard XII

Mathematics

Special Determinants

Express the f...

Question

Express the following matrix as the sum of a symmetric and a skew symmetric matrix:
$(i)$ $[\begin{matrix} 3 & 5 1 & - 1 \end{matrix}]$
$(i i)$ $⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦$
$(i i i)$ $⎡ ⎢ ⎣ \begin{matrix} 3 & 3 & - 1 - 2 & - 2 & 1 - 4 & - 5 & 2 \end{matrix} ⎤ ⎥ ⎦$
$(i v)$ $[\begin{matrix} 1 & 5 - 1 & 2 \end{matrix}]$

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Solution

$(i)$ Let $A = [\begin{matrix} 3 & 5 1 & - 1 \end{matrix}] \Rightarrow A^{'} = [\begin{matrix} 3 & 1 5 & - 1 \end{matrix}]$
Let $P = \frac{1}{2} (A + A^{'})$
$\Rightarrow P = \frac{1}{2} ([\begin{matrix} 3 & 5 1 & - 1 \end{matrix}] + [\begin{matrix} 3 & 1 5 & - 1 \end{matrix}])$
$\Rightarrow P = \frac{1}{2} [\begin{matrix} 6 & 6 6 & - 2 \end{matrix}]$
$\Rightarrow P = [\begin{matrix} 3 & 3 3 & - 1 \end{matrix}]$
$\Rightarrow P^{'} = [\begin{matrix} 3 & 3 3 & - 1 \end{matrix}] = P$
$∵ P^{'} = P$
$∴ P$ is a symmetric matrix.

Let $Q = \frac{1}{2} (A - A^{'})$
$\Rightarrow Q = \frac{1}{2} ([\begin{matrix} 3 & 5 1 & - 1 \end{matrix}] - [\begin{matrix} 3 & 1 5 & - 1 \end{matrix}])$
$\Rightarrow Q = \frac{1}{2} [\begin{matrix} 0 & 4 - 4 & 0 \end{matrix}]$
$\Rightarrow Q = [\begin{matrix} 0 & 2 - 2 & 0 \end{matrix}]$
$\Rightarrow Q^{'} = [\begin{matrix} 0 & - 2 2 & 0 \end{matrix}] = - [\begin{matrix} 0 & 2 - 2 & 0 \end{matrix}] = - Q$
$∵ Q^{'} = - Q$
$∴ Q$ is a skew symmetric matrix.
Now, $P + Q = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'})$
$\Rightarrow P + Q = A$
$∴ A$ is a sum of symmetric and skew symmetric matrix.
Hence
$\frac{[\begin{matrix} 3 & 5 1 & - 1 \end{matrix}]}{A Given Matrix}$ = $\frac{[\begin{matrix} 3 & 3 3 & - 1 \end{matrix}]}{P Symmetric Matrix}$ + $\frac{[\begin{matrix} 0 & 2 - 2 & 0 \end{matrix}]}{Q Skew Symmetric Matrix}$

$(i i)$ Let $A = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦$
$A^{'} = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦$
Let $P = \frac{1}{2} (A + A^{'})$
$\Rightarrow P = \frac{1}{2} ⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠$
$\Rightarrow P = \frac{1}{2} ⎡ ⎢ ⎣ \begin{matrix} 12 & - 4 & 4 - 4 & 6 & - 2 4 & - 2 & 6 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow P = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow P^{'} = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ = P$
$∵ P^{'} = P$
$∴ P$ is a symmetric matrix.

Let $Q = \frac{1}{2} (A - A^{'})$
$\Rightarrow Q = \frac{1}{2} ⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ - ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠$
$\Rightarrow Q = \frac{1}{2} ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow Q = ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow Q^{'} = ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦ = - ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦ = - Q$
$∵ Q^{'} = - Q$
$∴ Q$ is a skew symmetric matrix.
Now, $P + Q = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'})$
$\Rightarrow P + Q = A$
$∴ A$ is a sum of symmetric and skew symmetric matrix.
Hence
$\frac{⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦}{A Given Matrix}$ = $\frac{⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦}{P Symmetric Matrix}$ + $\frac{⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦}{Q Skew Symmetric Matrix}$

$(i i i)$ Let $A = ⎡ ⎢ ⎣ \begin{matrix} 3 & 3 & - 1 - 2 & - 2 & 1 - 4 & - 5 & 2 \end{matrix} ⎤ ⎥ ⎦$
$A^{'} = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 2 & - 4 3 & - 2 & - 5 - 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦$
Let $P = \frac{1}{2} (A + A^{'})$
$\Rightarrow P = \frac{1}{2} ⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} 3 & 3 & - 1 - 2 & - 2 & 1 - 4 & - 5 & 2 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 3 & - 2 & - 4 3 & - 2 & - 5 - 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠$
$\Rightarrow P = \frac{1}{2} ⎡ ⎢ ⎣ \begin{matrix} 6 & 1 & - 5 1 & - 4 & - 4 - 5 & - 4 & 4 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow P = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 3 & \frac{1}{2} & - \frac{5}{2} \frac{1}{2} & - 2 & - 2 - \frac{5}{2} & - 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
$\Rightarrow P = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 3 & \frac{1}{2} & - \frac{5}{2} \frac{1}{2} & - 2 & - 2 - \frac{5}{2} & - 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
$∵ P^{'} = P$
$∴ P$ is a symmetric matrix.

Let $Q = \frac{1}{2} (A - A^{'})$
$\Rightarrow Q = \frac{1}{2} ⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} 3 & 3 & - 1 - 2 & - 2 & 1 - 4 & - 5 & 2 \end{matrix} ⎤ ⎥ ⎦ - ⎡ ⎢ ⎣ \begin{matrix} 3 & - 2 & - 4 3 & - 2 & - 5 - 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠$
$\Rightarrow Q = \frac{1}{2} ⎡ ⎢ ⎣ \begin{matrix} 0 & 5 & 3 - 5 & 0 & 6 - 3 & - 6 & 0 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow Q = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & \frac{5}{2} & \frac{3}{2} - \frac{5}{2} & 0 & 3 - \frac{3}{2} & - 3 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
$\Rightarrow Q^{'} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & \frac{- 5}{2} & \frac{- 3}{2} \frac{5}{2} & 0 & - 3 \frac{3}{2} & 3 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = - ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & \frac{5}{2} & \frac{3}{2} - \frac{5}{2} & 0 & 3 - \frac{3}{2} & - 3 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = - Q$
$∵ Q^{'} = - Q$
$∴ Q$ is a skew symmetric matrix.
Now, $P + Q = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'})$
$\Rightarrow P + Q = A$
$∴ A$ sum of symmetric and skew symmetric matrix is $A$
Hence,
$\frac{⎡ ⎢ ⎣ \begin{matrix} 3 & 3 & - 1 - 2 & - 2 & 1 - 4 & - 5 & 2 \end{matrix} ⎤ ⎥ ⎦}{A Given Matrix}$ = $\frac{⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 3 & \frac{1}{2} & - \frac{5}{2} \frac{1}{2} & - 2 & - 2 - \frac{5}{2} & - 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦}{P Symmetric Matrix}$ + $\frac{⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & \frac{5}{2} & \frac{3}{2} - \frac{5}{2} & 0 & 3 - \frac{3}{2} & - 3 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦}{Q Skew Symmetric Matrix}$

$(i v)$ Let $A = [\begin{matrix} 1 & 5 - 1 & 2 \end{matrix}] \Rightarrow A^{'} = [\begin{matrix} 1 & - 1 5 & 2 \end{matrix}]$
Let $P = \frac{1}{2} (A + A^{'})$
$\Rightarrow P = \frac{1}{2} ([\begin{matrix} 1 & 5 - 1 & 2 \end{matrix}] + [\begin{matrix} 1 & - 1 5 & 2 \end{matrix}])$
$\Rightarrow P = \frac{1}{2} [\begin{matrix} 2 & 4 4 & 4 \end{matrix}]$
$\Rightarrow P = [\begin{matrix} 1 & 2 2 & 2 \end{matrix}]$
$\Rightarrow P^{'} = [\begin{matrix} 1 & 2 2 & 2 \end{matrix}] = P$
$∵ P^{'} = P$
$∴ P$ is a symmetric matrix.

Let $Q = \frac{1}{2} (A - A^{'})$
$\Rightarrow Q = \frac{1}{2} ([\begin{matrix} 1 & 5 - 1 & 2 \end{matrix}] - [\begin{matrix} 1 & - 1 5 & 2 \end{matrix}])$
$\Rightarrow Q = \frac{1}{2} [\begin{matrix} 0 & 6 - 6 & 0 \end{matrix}]$
$\Rightarrow Q = [\begin{matrix} 0 & 3 - 3 & 0 \end{matrix}]$
$\Rightarrow Q^{'} = [\begin{matrix} 0 & - 3 3 & 0 \end{matrix}] = - [\begin{matrix} 0 & 3 - 3 & 0 \end{matrix}] = - Q$
$∵ Q^{'} = - Q$
$∴ Q$ is a skew symmetric matrix.
Now, $P + Q = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'})$
$\Rightarrow P + Q = A$
$∴ A$ is a sum of symmetric and skew symmetric matrix.
Hence
$\frac{[\begin{matrix} 1 & 5 - 1 & 2 \end{matrix}]}{A Given Matrix}$ = $\frac{[\begin{matrix} 1 & 2 2 & 2 \end{matrix}]}{P Symmetric Matrix}$ + $\frac{[\begin{matrix} 0 & 3 - 3 & 0 \end{matrix}]}{Q Skew Symmetric Matrix}$

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Express the following matrix as the sum of a symmetric and a skew symmetric matrix: (i) [351−1] (ii) ⎡⎢⎣6−22−23−12−13⎤⎥⎦ (iii) ⎡⎢⎣33−1−2−21−4−52⎤⎥⎦ (iv) [15−12]