Find the eigenvalues of the matrix $A = \begin{pmatrix} 2 & 1 \ 1 & 2 \end{pmatrix}$.

Question

Accepted Answer

The eigenvalues $\lambda$ of a matrix $A$ are found by solving the characteristic equation:
$$|A - \lambda I| = 0$$

**Step 1: Set up the characteristic matrix**
$$A - \lambda I = \begin{pmatrix} 2-\lambda & 1 \ 1 & 2-\lambda \end{pmatrix}$|

**Step 2: Calculate the determinant and set to zero**
$$(2 - \lambda)(2 - \lambda) - (1)(1) = 0$$
$$(2 - \lambda)^2 - 1 = 0$$
$$(2 - \lambda)^2 = 1$$

**Step 3: Solve for $\lambda$**
$$2 - \lambda = \pm 1$$
- If $2 - \lambda = 1 \implies \lambda = 1$
- If $2 - \lambda = -1 \implies \lambda = 3$

Thus, the eigenvalues are **1 and 3**.

Find the eigenvalues of the matrix $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$. - DDCET Q&A