A $3 \times 3$ real symmetric matrix $M$ has eigenvalues $\lambda_1 = 2$, $\lambda_2 = -3$, and $\lambda_3 = 4$. What is the trace of the matrix $M^2$?

Answer Key:

29

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Eigenvalues of Matrix Powers:** If $\lambda$ is an eigenvalue of $M$, then $\lambda^k$ is an eigenvalue of $M^k$. 2. **Stage 2: Eigenvalues of $M^2$:** $\lambda_1^2 = 4$, $\lambda_2^2 = 9$, $\lambda_3^2 = 16$. 3. **Stage 3: Trace Evaluation:** $\text{Trace}(M^2) = 4 + 9 + 16 = 29$. 4. **Stage 4: Correct Value:** 29.

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