Matrix P: Symmetric Property Check

Matrix P is given by: $ P = \begin{bmatrix} -30 & 17 & 5 \\ 17 & 0 & -12 \\ 5 & -12 & 4 \end{bmatrix} $ A matrix is symmetric if it is equal to its transpose ($P = P^T$), meaning $P_{ij} = P_{ji}$ for all i, j.

• $P_{12} = 17$ and $P_{21} = 17$
• $P_{13} = 5$ and $P_{31} = 5$
• $P_{23} = -12$ and $P_{32} = -12$

Since $P_{ij} = P_{ji}$ for all elements, Matrix P is symmetric (Property 3).

Matrix Q: Trace Free Property Check

Matrix Q is given by: $ Q = \begin{bmatrix} 0 & 1 & 9 \\ 7 & 0 & 2 \\ 12 & 3 & 0 \end{bmatrix} $ A matrix is trace free if the sum of its diagonal elements (its trace) is zero.

Trace(Q) = $ Q_{11} + Q_{22} + Q_{33} $ Trace(Q) = $ 0 + 0 + 0 = 0 $

Since the trace is 0, Matrix Q is trace free (Property 4).

Matrix R: Singular Property Check

Matrix R is given by: $ R = \begin{bmatrix} 2/3 & 1/3 & 2/3 \\ 0 & 2/3 & -1/3 \\ 0 & 4/3 & -2/3 \end{bmatrix} $ A matrix is singular if its determinant is zero.

Calculate the determinant of R: det(R) = $ \frac{2}{3} \left( \frac{2}{3} \cdot \frac{-2}{3} - \left(-\frac{1}{3}\right) \cdot \frac{4}{3} \right) - \frac{1}{3} (0 \cdot \frac{-2}{3} - \left(-\frac{1}{3}\right) \cdot 0) + \frac{2}{3} (0 \cdot \frac{4}{3} - \frac{2}{3} \cdot 0) $ det(R) = $ \frac{2}{3} \left( \frac{-4}{9} + \frac{4}{9} \right) - \frac{1}{3}(0) + \frac{2}{3}(0) $ det(R) = $ \frac{2}{3} (0) - 0 + 0 = 0 $

Since det(R) = 0, Matrix R is singular (Property 1).

Matrix S: Triangular Property Check

Matrix S is given by: $ S = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix} $ A matrix is triangular if all the entries above or below the main diagonal are zero. This matrix has zeros below the main diagonal ($S_{21}=0, S_{31}=0, S_{32}=0$).

Therefore, Matrix S is triangular (Property 2).

Summary of Matches

The correct matches are:

This corresponds to the option P-3; Q-4; R-1; S-2.