A $3 \times 3$ matrix has trace equal to 5 and is symmetric. How many such matrices are possible?
Step-by-Step Solution
Key Concept: Symmetric matrices are determined by their diagonal and upper triangular elements only
Since the trace is equal to 5, the sum of diagonal elements is 5. The diagonal elements are 1, 2, 2 (in some order). For a symmetric matrix, we only need to specify the upper triangular part including the diagonal. With 3 diagonal elements and $\binom{3}{2} = 3$ independent off-diagonal elements, we have $3 \times 8 = 24$ possible matrices when accounting for the constraint that diagonal elements sum to 5.
Correct Answer: 24