Solved Examples: Finding Eigenvalues using det(A - λI) = 0.

📚 Quick Study Guide

🔢 Eigenvalue Definition: A scalar $\lambda$ is an eigenvalue of a square matrix $A$ if there exists a non-zero vector $v$ such that $Av = \lambda v$. 🔍 Characteristic Equation: The eigenvalues of a matrix $A$ are the solutions to the characteristic equation, given by $\det(A - \lambda I) = 0$, where $I$ is the identity matrix. 📐 Finding Eigenvalues:
1. Form the matrix $A - \lambda I$ by subtracting $\lambda$ from the diagonal elements of $A$.
2. Compute the determinant of $A - \lambda I$.
3. Set the determinant equal to zero and solve for $\lambda$. The solutions are the eigenvalues of $A$.
💡 Identity Matrix (I): An $n \times n$ identity matrix is a square matrix with 1s on the main diagonal and 0s everywhere else.

Practice Quiz

1. Which of the following equations is used to find eigenvalues?
   1. $\det(A + \lambda I) = 0$
   2. $\det(A - \lambda I) = 0$
   3. $\text{trace}(A) = 0$
   4. $\text{rank}(A) = 0$
2. Given the matrix $A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$, what is the characteristic equation?
   1. $(\lambda - 2)^2 - 1 = 0$
   2. $(\lambda + 2)^2 - 1 = 0$
   3. $(\lambda - 2)^2 + 1 = 0$
   4. $(\lambda + 2)^2 + 1 = 0$
3. For a $2 \times 2$ matrix $A$, if the eigenvalues are $\lambda_1 = 3$ and $\lambda_2 = 1$, what is the determinant of $A$?
   1. 2
   2. 3
   3. 4
   4. 3
4. What is the identity matrix ($I$) used for in the equation $\det(A - \lambda I) = 0$?
   1. To change the rank of $A$.
   2. To subtract $\lambda$ from all elements of $A$.
   3. To subtract $\lambda$ only from the diagonal elements of $A$.
   4. To add $\lambda$ to all elements of $A$.
5. If $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, what are the eigenvalues of $A$?
   1. $\lambda = 0, 1$
   2. $\lambda = 1, 1$
   3. $\lambda = 0, 0$
   4. $\lambda = -1, -1$
6. Suppose the characteristic equation of matrix $A$ is $(\lambda - 5)(\lambda + 2) = 0$. What are the eigenvalues of $A$?
   1. $\lambda = 5, -2$
   2. $\lambda = -5, 2$
   3. $\lambda = 5, 2$
   4. $\lambda = -5, -2$
7. For what type of matrix is it easiest to read the eigenvalues directly from the matrix?
   1. Symmetric Matrix
   2. Diagonal Matrix
   3. Invertible Matrix
   4. Singular Matrix