Discussion
If \(A = \begin{bmatrix} 5 & 2 & x \\ y & 2 & -3 \\ 4 & t & -7 \end{bmatrix}\) is a symmetric matrix, then the values of \(x, y\) and \(t\), respectively, are
Answer: B
\(x = 4\), \(y = 2\) and \(t = -3\).
A symmetric matrix equals its transpose, so the entry in row \(i\), column \(j\) equals the entry in row \(j\), column \(i\):
\[ \begin{aligned} a_{13} &= a_{31}: \quad x = 4 \\ a_{21} &= a_{12}: \quad y = 2 \\ a_{32} &= a_{23}: \quad t = -3 \end{aligned} \]
4,2,3 drops the sign of \(-3\); 4,2,\(-7\) wrongly takes \(t\) from the diagonal entry \(-7\); 2,4,\(-7\) swaps \(x\) and \(y\) as well; 4,3,2 has no basis in the matrix.
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