Question

A matrix P is decomposed into its symmetric part S and skew symmetric part V.

If \({\rm{S}} = \left( {\begin{array}{*{20}{c}} { - 4}&4&2\\ 4&3&{7/2}\\ 2&{7/2}&2 \end{array}} \right),{\rm{\;V}} = \left( {\begin{array}{*{20}{c}} 0&{ - 2}&3\\ 2&0&{7/2}\\ { - 3}&{ - 7/2}&0 \end{array}} \right)\)

then matrix P is

1. \(\left( {\begin{array}{*{20}{c}} { - 4}&6&{ - 1}\\ 2&3&0\\ 5&7&2 \end{array}} \right)\)
2. \(\left( {\begin{array}{*{20}{c}} { - 4}&2&5\\ 6&3&7\\ { - 1}&0&2 \end{array}} \right)\)
3. \(\left( {\begin{array}{*{20}{c}} 4&{ - 6}&1\\ { - 2}&{ - 3}&0\\ { - 5}&{ - 7}&{ - 2} \end{array}} \right)\)
4. \(\left( {\begin{array}{*{20}{c}} { - 2}&{9/2}&{ - 1}\\ { - 1}&{81/4}&{11}\\ { - 2}&{45/2}&{73/4} \end{array}} \right)\)

Answer 1

Correct Answer - Option 2 : \(\left( {\begin{array}{*{20}{c}} { - 4}&2&5\\ 6&3&7\\ { - 1}&0&2 \end{array}} \right)\)

Concept:

Every square matrix is expressed as the sum of symmetric and skew-symmetric matrix. Here, S is symmetric matrix and V is skew-symmetric matrix.

∴ P = S + V

Calculation:

\(P = \;\left[ {\begin{array}{*{20}{c}} { - 4}&4&2\\ 4&3&{\frac{7}{2}}\\ 2&{\frac{7}{2}}&2 \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0&{ - 2}&3\\ 2&0&{\frac{7}{2}}\\ { - 3}&{ - \frac{7}{2}}&0 \end{array}} \right]\)

\(\therefore P = \;\left[ {\begin{array}{*{20}{c}} { - 4}&2&5\\ 6&3&7\\ { - 1}&0&2 \end{array}} \right]\)