Let A be the matrix \( \begin{bmatrix}3 & -4\\1 & -1\end{bmatrix} \). If A is expressed as the sum of P and Q, where P is a symmetric and Q is a skew symmetric matrix, then which of the following options define P and Q correctly ?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \mathrm{P}=\begin{bmatrix}3 & -\displaystyle\frac{3}{2}\\-\displaystyle\frac{3}{2} & -1\end{bmatrix} \), \( \mathrm{Q}=\begin{bmatrix}0 & -\displaystyle\frac{5}{2}\\\displaystyle\frac{5}{2} & 0\end{bmatrix} \)
Every square matrix splits as \( \mathrm{A}=\frac{1}{2}(\mathrm{A}+\mathrm{A}^T)+\frac{1}{2}(\mathrm{A}-\mathrm{A}^T) \), where the first part is symmetric and the second is skew symmetric.
\( \displaystyle \mathrm{A}^T=\begin{bmatrix}3 & 1\\-4 & -1\end{bmatrix} \)
\( \displaystyle \mathrm{P}=\frac{1}{2}(\mathrm{A}+\mathrm{A}^T) \)
\( \displaystyle =\frac{1}{2}\begin{bmatrix}6 & -3\\-3 & -2\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix}3 & -\frac{3}{2}\\-\frac{3}{2} & -1\end{bmatrix} \)
\( \displaystyle \mathrm{Q}=\frac{1}{2}(\mathrm{A}-\mathrm{A}^T) \)
\( \displaystyle =\frac{1}{2}\begin{bmatrix}0 & -5\\5 & 0\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix}0 & -\frac{5}{2}\\\frac{5}{2} & 0\end{bmatrix} \)
Check: \( \mathrm{P}+\mathrm{Q}=\begin{bmatrix}3 & -4\\1 & -1\end{bmatrix}=\mathrm{A} \).
Shortcut: Test the options without calculating: Q must have zeros on its diagonal and \( q_{21}=-q_{12} \), and P must have \( p_{12}=p_{21} \). Option (2) and Option (4) fail because their Q has non-zero diagonal entries, and Option (3) fails because its P is not symmetric. Only Option (1) is left.