Step 1: Identify the Base Matrix:
Let $B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. We need the sum of eigenvalues of $A = B^2$.
Step 2: Find Trace and Determinant of $B$
Sum of eigenvalues of $B$ ($\lambda_1 + \lambda_2$) = Trace = $1 + 4 = 5$. \\
Product of eigenvalues of $B$ ($\lambda_1 \lambda_2$) = Det = $(1 \times 4) - (3 \times 2) = -2$.
Step 3: Apply the Power Property:
If $\lambda$ is an eigenvalue of $B$, then $\lambda^2$ is an eigenvalue of $B^2$.
Sum of eigenvalues of $A = \lambda_1^2 + \lambda_2^2$.
Step 4: Use Algebraic Identity
$\lambda_1^2 + \lambda_2^2 = (\lambda_1 + \lambda_2)^2 - 2(\lambda_1 \lambda_2)$
Substitute the values from Step 2: \\
Sum $= (5)^2 - 2(-2)$ \\
Sum $= 25 + 4 = 29$.
