Find x, y, z, t if A is a Symmetric Matrix

Given:

\[ A = \begin{bmatrix} 5 & 2 & x \\ y & z & -3 \\ 4 & t & -7 \end{bmatrix} \]

Condition for Symmetric Matrix:

\[ A = A^T \]

Step 1: Compare Corresponding Elements

\[ a_{12} = a_{21} \Rightarrow 2 = y \Rightarrow y = 2 \]

\[ a_{13} = a_{31} \Rightarrow x = 4 \]

\[ a_{23} = a_{32} \Rightarrow -3 = t \Rightarrow t = -3 \]

\[ a_{22} = a_{22} \Rightarrow z = z \quad (\text{no restriction}) \]

Final Answer:

\[ x = 4, \quad y = 2, \quad t = -3, \quad z \text{ is any real number} \]

Conclusion:

The matrix becomes symmetric when corresponding elements are equal across the main diagonal.