If \(A = \left[ {\begin{array}{*{20}{c}} { - \;1}&5&1\\ 2&3&4\\ 7&0&9 \end{array}} \right] = \frac{1}{2} \cdot (P + Q)\) where P is symmetric and Q is skew symmetric matrix then P and Q are ?

Concept:

• Symmetric Matrix: Any real square matrix A = (aij) is said to be symmetric matrix if and only if aij = aji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = A’ then A is said to be a symmetric matrix.
• Skew-symmetric Matrix: Any real square matrix A = (aij) is said to be skew-symmetric matrix if and only if aij = - aji, ∀ i and j or in other words we can say that if A is a real square matrix such that A =- A’ then A is said to be a skew-symmetric matrix.

• Any real square matrix say A, can be expressed as sum of symmetric and skew-symmetric matrix.

  i.e \(A = \frac{1}{2}\;\left( {A + A'} \right) + \frac{1}{2}\;\left( {A - A'} \right)\) where A + A’ is symmetric and A – A’ is skew-symmetric matrix.

Calculation:

Given: \(A = \left[ {\begin{array}{*{20}{c}} { - \;1}&5&1\\ 2&3&4\\ 7&0&9 \end{array}} \right] = \frac{1}{2} \cdot (P + Q)\) where P is symmetric and Q is skew symmetric matrix

Here we have to find the matrix P and Q

As we know that, any square matrix can be be expressed as sum of symmetric and skew-symmetric matrix.

i.e If A is a square matrix then A can be expressed as: \(A = \frac{1}{2}\;\left( {A + A'} \right) + \frac{1}{2}\;\left( {A - A'} \right)\) where A + A’ is symmetric and A – A’ is skew-symmetric matrix.

By comparing \(A = \left[ {\begin{array}{*{20}{c}} { - \;1}&5&1\\ 2&3&4\\ 7&0&9 \end{array}} \right] = \frac{1}{2} \cdot (P + Q)\) with \(A = \frac{1}{2}\;\left( {A + A'} \right) + \frac{1}{2}\;\left( {A - A'} \right)\) we get,

⇒ P = A + A' and Q = A - A'

\(A = \left[ {\begin{array}{*{20}{c}} { - \;1}&5&1\\ 2&3&4\\ 7&0&9 \end{array}} \right]\;and\;A' = \left[ {\begin{array}{*{20}{c}} { - \;1}&2&7\\ 5&3&0\\ 1&4&9 \end{array}} \right]\)

⇒ \(P = \;\left[ {\begin{array}{*{20}{c}} { - \;1}&5&1\\ 2&3&4\\ 7&0&9 \end{array}} \right] + \;\left[ {\begin{array}{*{20}{c}} { - \;1}&2&7\\ 5&3&0\\ 1&4&9 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - \;2}&7&8\\ 7&6&4\\ 8&4&{18} \end{array}} \right]\)

Similarly, \(Q = \;\left[ {\begin{array}{*{20}{c}} { - \;1}&5&1\\ 2&3&4\\ 7&0&9 \end{array}} \right] - \;\left[ {\begin{array}{*{20}{c}} { - \;1}&2&7\\ 5&3&0\\ 1&4&9 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&3&{ - \;6}\\ { - \;3}&0&4\\ 6&{ - \;4}&0 \end{array}} \right]\)

Hence, \(P = \;\left[ {\begin{array}{*{20}{c}} { - \;2}&7&8\\ 7&6&4\\ 8&4&{18} \end{array}} \right]\;and\;Q = \;\left[ {\begin{array}{*{20}{c}} 0&3&{ - \;6}\\ { - \;3}&0&4\\ 6&{ - \;4}&0 \end{array}} \right]\)