Question

[tex] Express \left[\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right] as a sum of a symmetric matrix and a skew-symmetric matrix. [\tex]

Answer 1

Answer:

$A=\left[\begin{array}{ccc}1&\frac{1}{2} &3\\\frac{1}{2} &1&1\\3&1&5\end{array}\right]+\left[\begin{array}{ccc}0&\dfrac{-3}{2}&-1\\\dfrac{3}{2}&0&2\\1&-2&0\end{array}\right]$

Step-by-step explanation:

$Given,\\A=\left[\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right]\\Transpose\ of\ A, A^{T} =\left[\begin{array}{ccc}1&2&4\\-1&1&-1\\2&3&5\end{array}\right]$

$symmetrix\ matrix =\frac{A+A^{T} }{2}\\=\frac{1}{2}{\left[\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right] +\left[\begin{array}{ccc}1&2&4\\-1&1&-1\\2&3&5\end{array}\right]} \\=\left[\begin{array}{ccc}1&\frac{1}{2} &3\\\frac{1}{2} &1&1\\3&1&5\end{array}\right]$

$skew-symmetrix\ matrix =\frac{A-A^{T} }{2}\\=\dfrac{1}{2}(\left[\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right]-\left[\begin{array}{ccc}1&2&4\\-1&1&-1\\2&3&5\end{array}\right]\\=\left[\begin{array}{ccc}0&\dfrac{-3}{2}&-1\\\dfrac{3}{2}&0&2\\1&-2&0\end{array}\right]$

hence ,A as a sum of symmetrx matrix and skew-symmetrix matrix$A=\left[\begin{array}{ccc}1&\frac{1}{2} &3\\\frac{1}{2} &1&1\\3&1&5\end{array}\right]+\left[\begin{array}{ccc}0&\dfrac{-3}{2}&-1\\\dfrac{3}{2}&0&2\\1&-2&0\end{array}\right]$