Question

The sum of the eigenvalues of the matrix 
\( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^2 \) 
is _____________ (rounded off to the nearest integer). 

Hint:

The trace of a matrix equals the sum of its eigenvalues. For powers of matrices, calculate carefully using the characteristic equation.

Answer 1

Correct Answer: 29

Given:

\( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^2 \)

---

Step 1: Compute \( A^2 \)

First compute the matrix square:

\( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)

\( A^2 = A \cdot A = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix} \)

---

Step 2: Find eigenvalues of \( A^2 \)

The characteristic equation is:

\( |A^2 - \lambda I| = 0 \)

\( \lambda^2 - 29\lambda + 154 = 0 \)

Solving this quadratic equation:

\( \lambda_1 = 28.8615, \quad \lambda_2 = 0.1385 \)

---

Step 3: Sum of eigenvalues

\( \lambda_1 + \lambda_2 = 28.8615 + 0.1385 = 29 \)

---

Final Answer:

29