Discussion
If \(\mathbf{A} = \begin{bmatrix} 10 & 2k + 5 \\ 3k - 3 & k + 5 \end{bmatrix}\) is a symmetric matrix, the value of k is ___________.
Answer: A
The value is \(k = 8\).
A matrix is symmetric when it equals its transpose, so the off-diagonal entries must be equal:
\[ 2k + 5 = 3k - 3 \Rightarrow k = 8 \]
Then both off-diagonal entries are 21, and the diagonal entries (10 and 13) are unrestricted.
\(k = 5\) gives 15 and 12, \(k = -0.4\) gives 4.2 and -4.2 (equal and opposite, a skew-type condition on those entries), and the surd value does not make the two entries equal either.
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