📘 Question
Matrix \[ A = \begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix} \] is symmetric. Find the values of \(a\) and \(b\).
✏️ Step-by-Step Solution
Step 1: Use symmetry condition
For a symmetric matrix:
\[
A = A^T
\]
So, corresponding elements must be equal:
- \(a_{12} = a_{21} \Rightarrow 2b = 3\)
- \(a_{13} = a_{31} \Rightarrow -2 = 3a\)
Step 2: Solve equations
From \(2b = 3\):
\[
b = \frac{3}{2}
\]
From \(-2 = 3a\):
\[
a = -\frac{2}{3}
\]
✅ Final Answer
\[
\boxed{a = -\frac{2}{3}, \quad b = \frac{3}{2}}
\]
💡 Key Concept
A matrix is symmetric if \(A = A^T\). This means elements across the diagonal are equal:
- \(a_{ij} = a_{ji}\)
Use this property to form equations and solve unknown values.