To compute the eigenvalues of a matrix $A$, follow these step-by-step instructions:

Step 1: Understand the Eigenvalue Equation

Eigenvalues are found using the characteristic equation: $\det(A - \lambda I) = 0$ where:

• $A$ is the given square matrix.
• $\lambda$ represents the eigenvalues.
• $I$ is the identity matrix of the same dimension as $A$.

Step 2: Subtract $\lambda I$ from $A$

Replace the diagonal elements of $A$ with $a_{ii} - \lambda$. For example: $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad \lambda I = \begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix}$ $A - \lambda I = \begin{bmatrix} a - \lambda & b \\ c & d - \lambda \end{bmatrix}$

Step 3: Compute the Determinant of $A - \lambda I$

Find the determinant of $A - \lambda I$. For a $2 \times 2$ matrix: $\det(A - \lambda I) = (a - \lambda)(d - \lambda) - bc$ Simplify this determinant into a polynomial equation in $\lambda$.

For larger matrices (e.g., $3 \times 3$ or higher), use cofactor expansion or row reduction to compute the determinant.

Step 4: Solve the Characteristic Equation

Set the determinant equal to zero: $\det(A - \lambda I) = 0$ Solve for $\lambda$. This might involve factoring, solving quadratic equations, or finding roots of higher-degree polynomials.

Example for a $2 \times 2$ Matrix

Let: $A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$

1. Compute $A - \lambda I$: $A - \lambda I = \begin{bmatrix} 4-\lambda & 1 \\ 2 & 3-\lambda \end{bmatrix}$

2. Calculate the determinant: $\det(A - \lambda I) = (4-\lambda)(3-\lambda) - (2)(1)$ Expand: $\det(A - \lambda I) = 12 - 4\lambda - 3\lambda + \lambda^2 - 2$ $= \lambda^2 - 7\lambda + 10$

3. Solve the characteristic equation: $\lambda^2 - 7\lambda + 10 = 0$ Factorize: $(\lambda - 5)(\lambda - 2) = 0$ So, $\lambda = 5$ and $\lambda = 2$.

Step 5: Verify

Substitute $\lambda = 5$ and $\lambda = 2$ into $\det(A - \lambda I)$ to ensure they satisfy the equation.

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Would you like to solve a specific matrix? If yes, share the matrix, and I’ll walk you through the process.

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5 Related Questions:

1. How do eigenvalues relate to eigenvectors?
2. How can eigenvalues be interpreted in real-world applications?
3. What happens if a matrix is singular (determinant = 0)?
4. How do eigenvalues change if the matrix is scaled by a constant?
5. Can eigenvalues be complex numbers?

Tip:

Always double-check your determinant calculations, especially for larger matrices, as small errors can lead to incorrect eigenvalues.