Question

If a matrix \(M\) is defined as \(M=\begin{bmatrix}10 & 6 \\ 6 & 10\end{bmatrix}\), the sum of all the eigenvalues of \(M^3\) is equal to ________________ (in integer).

Hint:

Sum of eigenvalues of a matrix equals its trace.
For powers of matrices, eigenvalues are raised to that power.

Answer 1

Correct Answer: 4160

Step 1: Find eigenvalues of \(M\)
For a \(2\times 2\) symmetric matrix \(\begin{bmatrix}a & b
b & a\end{bmatrix}\), eigenvalues are \(a+b\) and \(a-b\). Here \(a=10,\; b=6\). So \[ \lambda_1=10+6=16, \quad \lambda_2=10-6=4. \] Step 2: Eigenvalues of \(M^3\)
If \(\lambda\) is an eigenvalue of \(M\), then \(\lambda^3\) is an eigenvalue of \(M^3\). Thus eigenvalues of \(M^3\) are: \[ 16^3=4096,\quad 4^3=64. \] Step 3: Sum of eigenvalues
\[ 4096+64=4160. \] So the sum of all eigenvalues of \(M^3\) is \(\boxed{4160}\).