GATE CH 2023 | Question: 47

Question

If a matrix $M$ is defined as $M=\left[\begin{array}{cc}10 & 6 \\ 6 & 10\end{array}\right]$, the sum of all the eigenvalues of $M^3$ is equal to______________(in integer).

Answer 1

Step 1: Identify Properties of Matrix  $M$
Sum of eigenvalues ($\lambda_1 + \lambda_2$) = Trace = $10 + 10 = 20$.
Product of eigenvalues ($\lambda_1 \lambda_2$) = Det = $(10 \times 10) - (6 \times 6) = 64$.

Step 2: Solve for Individual Eigenvalues:
Characteristic Equation: $\lambda^2 - 20\lambda + 64 = 0$
$(\lambda - 16)(\lambda - 4) = 0 \implies \lambda_1 = 16, \lambda_2 = 4$.

Step 3: Calculate Sum of Eigenvalues for  $M^3$
Sum $= \lambda_1^3 + \lambda_2^3 = (16)^3 + (4)^3$
Sum $= 4096 + 64 = 4160$.