Given the $2 \times 2$ invertible matrix $A = \begin{pmatrix} 3 & 4 \\ 1 & 3 \end{pmatrix}$, what is the determinant of the inverse matrix, $\det(A^{-1})$?

Answer Key:

0.2

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Determinant of A:** $\det(A) = ad - bc = (3)(3) - (4)(1) = 5$. 2. **Stage 2: Determinant Identity:** By matrix algebra properties, $\det(A^{-1}) = \frac{1}{\det(A)}$. 3. **Stage 3: Computation:** $\det(A^{-1}) = 1/5 = 0.2$. 4. **Stage 4: Final Value:** 0.2 (accepted range 0.19 to 0.21).

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