Solve Linear Algebra Problems: Matrix Operations, Determinants, Eigenvalues, and Decompositions with Step-by-Step Solutions
Written by Dr. Sarah Chen, PhD in Applied Mathematics | Reviewed by Prof. Michael Rodriguez, Linear Algebra Expert
โ Mathematically Verified - Updated Jan 4, 2026
Learn matrix operations with detailed, step-by-step solutions. Each example demonstrates key concepts in linear algebra and shows how our calculator solves them.
Multiply $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$:
Calculate $\det(A)$ for $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix}$:
Find eigenvalues and eigenvectors of $A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$:
Matrices are rectangular arrays of numbers that represent linear transformations between vector spaces. They are fundamental to linear algebra and have applications in virtually every field of science, engineering, and mathematics.
For $A \in \mathbb{R}^{m \times n}$ and $B \in \mathbb{R}^{n \times p}$, the product $C = AB \in \mathbb{R}^{m \times p}$ has entries:
$c_{ij} = \sum_{k=1}^n a_{ik}b_{kj}$
Matrix multiplication is associative but not commutative.
The determinant $\det(A)$ measures how a linear transformation changes volume. For $2\times2$:
$\det\begin{bmatrix}a&b\\c&d\end{bmatrix} = ad - bc$
A matrix is invertible iff $\det(A) \neq 0$.
$\lambda$ is eigenvalue and $v$ eigenvector if $Av = \lambda v$. Eigenvalues solve $\det(A - \lambda I) = 0$.
Eigen decomposition: $A = PDP^{-1}$ where $D$ diagonal of eigenvalues.
Matrix decompositions break matrices into simpler components, revealing their structure and enabling efficient computation:
| Decomposition | Formula | Applications |
|---|---|---|
| LU Decomposition | $A = LU$ where $L$ lower triangular, $U$ upper triangular | Solving linear systems, matrix inversion |
| QR Decomposition | $A = QR$ where $Q$ orthogonal, $R$ upper triangular | Least squares problems, eigenvalue algorithms |
| Cholesky Decomposition | $A = LL^T$ for positive definite $A$ | Numerical optimization, Monte Carlo methods |
| Singular Value Decomposition (SVD) | $A = U\Sigma V^T$ where $U,V$ orthogonal, $\Sigma$ diagonal | Principal Component Analysis, image compression |
| Eigen Decomposition | $A = PDP^{-1}$ where $D$ diagonal of eigenvalues | Matrix powers, differential equations, Markov chains |
As covered in our positive definite matrices guide, a symmetric matrix $A$ is positive definite if $x^TAx > 0$ for all nonzero $x$. This has important implications:
Matrices represent transformations: rotation, scaling, translation, projection. 3D graphics uses $4\times4$ homogeneous coordinates.
Quantum states are vectors in Hilbert spaces, observables are Hermitian matrices, measurements are eigenvalues.
Neural networks use weight matrices. PCA uses eigenvalue decomposition of covariance matrices. Recommender systems use matrix factorization.
Adjacency matrices represent graphs. Eigenvalues reveal connectivity. Google's PageRank uses eigenvalue problem.
Explore our comprehensive guides to deepen your understanding of linear algebra and matrix theory.
Solve limits step-by-step including the viral inner product limit problem. Essential for calculus and analysis.
Compute vector operations: dot product, cross product, magnitude, angle, and projections with step-by-step solutions.
Find eigenvalues and eigenvectors of matrices with detailed step-by-step solutions and geometric interpretations.