Question

The sum of the eigenvalues 𝜆₁ and 𝜆₂ of matrix 𝐵 = 𝐼+𝐴+𝐴² , where
\( \left[ {\begin{array}{cc}  2&1 \\     -0.5 &0.5 \\   \end{array} } \right]\)
 is_____ (rounded off to two decimal places).

Answer 1

Correct Answer: 7.7 - 7.8

We need to find the sum of the eigenvalues \( \lambda_1 \) and \( \lambda_2 \) of the matrix \( B = I + A + A^2 \), where the matrix \( A \) is:

\[ A = \begin{bmatrix} 2 & -0.5 \\ 1 & 0.5 \end{bmatrix} \] 

1. **Calculate \( A^2 \)**:

First, we compute \( A^2 \), which is the square of the matrix \( A \): \[ A^2 = A \times A = \begin{bmatrix} 2 & -0.5 \\ 1 & 0.5 \end{bmatrix} \times \begin{bmatrix} 2 & -0.5 \\ 1 & 0.5 \end{bmatrix} \] Performing matrix multiplication: \[ A^2 = \begin{bmatrix} 3.5 & -1.25 \\ 2.5 & -0.25 \end{bmatrix} \]

2. **Calculate matrix \( B \)**:

Now, we compute \( B = I + A + A^2 \), where \( I \) is the identity matrix:

\[ B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 2 & -0.5 \\ 1 & 0.5 \end{bmatrix} + \begin{bmatrix} 3.5 & -1.25 \\ 2.5 & -0.25 \end{bmatrix} \] Simplifying: \[ B = \begin{bmatrix} 6.5 & -1.75 \\ 3.5 & 1.25 \end{bmatrix} \]

3. **Find the eigenvalues of \( B \)**:

To find the eigenvalues of \( B \), we use the characteristic equation:

\[ \text{det}(B - \lambda I) = 0 \] The characteristic equation is: \[ \text{det} \begin{bmatrix} 6.5 - \lambda & -1.75 \\ 3.5 & 1.25 - \lambda \end{bmatrix} = 0 \] Expanding the determinant: \[ (6.5 - \lambda)(1.25 - \lambda) - (-1.75)(3.5) = 0 \] Simplifying: \[ \lambda^2 - 7.75\lambda + 14.25 = 0 \] Using the quadratic formula: \[ \lambda = \frac{-(-7.75) \pm \sqrt{(-7.75)^2 - 4(1)(14.25)}}{2(1)} \] \[ \lambda = \frac{7.75 \pm \sqrt{60.0625 - 57}}{2} \] \[ \lambda = \frac{7.75 \pm \sqrt{3.0625}}{2} \] \[ \lambda = \frac{7.75 \pm 1.75}{2} \] The eigenvalues are: \[ \lambda_1 = 4.75 \quad \text{and} \quad \lambda_2 = 3 \]

4. **Sum of the Eigenvalues**:

The sum of the eigenvalues \( \lambda_1 + \lambda_2 \) is:

\[ \lambda_1 + \lambda_2 = 4.75 + 3 = 7.75 \]

Thus, the sum of the eigenvalues is approximately \( 7.75 \), which rounds to \( 7.70 \) to \( 7.80 \).