Welcome to our collection of practice problems in linear algebra. These problems are designed to help you deepen your understanding of the subject. We hope you find them challenging and enlightening.
Matrix Operations
Problem: If ( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} ) and ( B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} ), find ( A \cdot B ).
Problem: Determine if the matrix ( A = \begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix} ) is invertible. If it is, find its inverse.
Systems of Linear Equations
Problem: Solve the following system of equations: [ \begin{align*} 2x + 3y &= 8 \ 4x - y &= 5 \end{align*} ]
Problem: Use Gaussian elimination to find the solution set of the system: [ \begin{align*} 3x + 2y + z &= 6 \ x - 4y + 2z &= -7 \ 2x + y - 3z &= 1 \end{align*} ]
Eigenvalues and Eigenvectors
Problem: Find the eigenvalues and eigenvectors of the matrix ( A = \begin{bmatrix} 4 & -2 \ 2 & 1 \end{bmatrix} ).
Problem: Determine if the matrix ( A = \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} ) is diagonalizable. If it is, find a diagonalizing matrix.
For more in-depth explanations and further practice, be sure to check out our Linear Algebra Guide.
Visualizing Vectors
Understanding vectors is crucial in linear algebra. Here's a visual representation of a vector in 3D space.