Find the eigenvalues and eigenvectors of the following matrices. Do some problems by hand to be sure you understand what the process means. Then check your results by computer. $$\left(\begin{array}{rrr} 1 & 1 & -1 \\ 1 & 1 & 1 \\ -1 & 1 & -1 \end{array}\right)$$

Matrix algebra is a foundation of linear algebra. Matrices are rectangular arrays of numbers, and they provide a compact way to handle systems of linear equations and transformations. Understanding matrix operations is crucial for solving problems involving eigenvalues and eigenvectors.

Some basic operations include:

• Matrix Addition: Adding corresponding elements of two matrices.
• Matrix Multiplication: Combining elements according to specific rules, leading to a new matrix.
• Determinant: A scalar value that describes some properties of the matrix, like invertibility.
• Inverse: If a matrix can be multiplied by another to yield the identity matrix, it has an inverse.

For eigenvalues and eigenvectors, we often use a determinant and finding inverses. Knowing these operations helps tackle more complex problems, such as the one above.

The characteristic equation is derived from the matrix and is key to finding eigenvalues. It's generally obtained from the equation: \[ \text{det}(A - \text{I} \times \text{λ}) = 0 \] Here, A is the original matrix, I is the identity matrix, and λ (lambda) represents the eigenvalues.

This determinant results in a polynomial equation in terms of λ. Solving this equation is essential since its roots give us the eigenvalues. Each eigenvalue reveals vital information about the matrix, like potential invariant transformations.

For instance, in the given problem, we set up: \[ \text{det}\begin{pmatrix} 1-\text{λ} & 1 & -1 \ 1 & 1-\text{λ} & 1 \ -1 & 1 & -1-\text{λ} \ \end{pmatrix} = 0 \] Simplifying this gives us a polynomial whose roots are the eigenvalues.

Cofactor expansion is a technique used to simplify the computation of a determinant. It involves breaking down a larger matrix into smaller chunks. This method is particularly handy for matrices larger than 2x2.

To perform a cofactor expansion, choose a row or column and take the sum of products of elements and their cofactors. For example, in the given exercise, we expand along the first row of our 3x3 matrix:

\[ (1-\text{λ}) \text{det}\begin{pmatrix} 1-λ & 1 \ 1 & -1-λ \ \end{pmatrix} - 1 \text{det}\begin{pmatrix} 1 & 1 \ -1 & -1-λ \ \end{pmatrix} + (-1) \text{det}\begin{pmatrix} 1 & 1-λ \ -1 & 1 \ \end{pmatrix} \] This exposes smaller determinants that are easier to handle.

Cofactor expansion always improves understanding of each matrix's properties and simplifies the characteristic equation.

Linear algebra is the branch of mathematics concerning linear equations and their representations. Here, we'll focus on key techniques related to eigenvalues and eigenvectors.

• Eigenvalues and Eigenvectors: These are scalar and vector combinations where a given matrix acts only as a scalar multiplier. This reveals invariant directions under the transformation.
• Diagonalization: Transforming a matrix into a diagonal form using its eigenvalues and eigenvectors.
• Solving Systems: Using matrices and their properties (like eigenvalues) simplifies solving linear systems.
• Transformation Geometries: Understanding how linear transformations function via matrices.

In the given exercise, we use these techniques to uncover the eigenvalues and corresponding eigenvectors. We initially compute the characteristic equation, use cofactor expansion to simplify determinants, and find solutions to eigenvalue-based equations. Verifying with computing tools ensures accuracy and solidifies understanding.