Comprehensive suite of matrix calculators for linear algebra operations. Calculate matrix determinants, inverses, perform addition, multiplication, and find eigenvalues instantly.
Free online matrix tools designed for students, engineers, and mathematicians. Solve complex linear algebra problems with step-by-step solutions and detailed explanations.
Professional-grade matrix calculations with support for various matrix sizes and operations. Perfect for homework, engineering projects, and mathematical research.
Enter matrix values row by row. Separate numbers with spaces. Example for 2x2 matrix:1 23 4
Enter matrix values to calculate determinant
Select the matrix operation you need (determinant, inverse, addition, multiplication, or eigenvalues)
Enter your matrix dimensions (rows × columns) or use preset sizes for common operations
Input the matrix elements in the provided grid or textarea
For operations requiring multiple matrices, enter all necessary matrices
Click 'Calculate' to perform the selected matrix operation
View the detailed result with step-by-step solution process
Export or copy the results for your assignments or documentation
Calculate the determinant of a 3×3 transformation matrix to check if it preserves volume
Solve simultaneous equations using matrix inverse for electrical circuit analysis
Matrix multiplication to calculate inter-industry economic dependencies
Eigenvalue calculation to analyze long-term population growth rates
Matrix operations for image rotation and scaling transformations
Solve systems of linear equations, analyze structural mechanics, perform circuit analysis, and solve differential equations. Essential for mechanical, electrical, and civil engineering applications.
Perform 3D transformations, calculate object rotations, implement perspective projections, and create realistic animations. Matrix operations are fundamental to modern computer graphics.
Implement PCA for dimensionality reduction, calculate covariance matrices, solve linear regression problems, and perform neural network computations. Critical for data analysis and AI applications.
Model economic systems, analyze input-output relationships, optimize investment portfolios, and calculate risk matrices. Used for economic forecasting and financial modeling.
Solve quantum mechanics problems, analyze wave functions, perform coordinate transformations, and model physical systems. Essential for theoretical and applied physics research.
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are fundamental tools in linear algebra, providing a compact way to represent and manipulate linear systems of equations. Each element in a matrix is identified by its position (i,j) where i represents the row and j represents the column. Matrix operations follow specific rules that make them powerful for solving complex mathematical problems.
The determinant is a scalar value calculated from a square matrix that provides important information about the matrix's properties. A non-zero determinant indicates that the matrix is invertible and the associated system of equations has a unique solution. The determinant can be calculated using various methods including cofactor expansion, row reduction, or the Leibniz formula. For 2×2 matrices, det(A) = ad - bc, while larger matrices require more complex calculations.
Matrix multiplication is not commutative, meaning A × B ≠ B × A in general. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix has dimensions equal to the number of rows of the first matrix by the number of columns of the second matrix. This operation is fundamental in transformations and solving systems of equations.
Eigenvalues and eigenvectors represent special properties of square matrices. An eigenvector v of matrix A satisfies Av = λv, where λ is the corresponding eigenvalue. These concepts are crucial in many applications including principal component analysis, vibration analysis, and quantum mechanics. Finding eigenvalues involves solving the characteristic equation det(A - λI) = 0.
Incorrect matrix dimensions for multiplication
Always verify that the number of columns in the first matrix equals the number of rows in the second matrix. For A(m×n) × B(n×p), the result will be C(m×p).
Attempting to find inverse of singular matrices
Check that the determinant is non-zero before calculating the inverse. A zero determinant indicates the matrix is singular (non-invertible).
Sign errors in cofactor expansion
Remember the alternating sign pattern in cofactor expansion: (+, -, +) in the first row, (-, +, -) in the second row, and so on. Use the formula (-1)^(i+j) for consistency.
Confusing matrix addition with element-wise multiplication
Matrix addition adds corresponding elements, while matrix multiplication involves dot products of rows and columns. Don't confuse these operations or mix up their formulas.
Forgetting to verify eigenvalue calculations
Always verify your eigenvalues by checking if det(A - λI) = 0. Also confirm that Av = λv holds for each eigenvalue-eigenvector pair you calculate.
Rounding errors in complex calculations
Maintain precision throughout calculations, especially when working with large matrices or near-zero values. Use exact fractions when possible and only round at the final step.
Our matrix calculators support rectangular and square matrices up to 10×10 dimensions. They handle real numbers, complex numbers, and can perform all basic operations including determinant, inverse, multiplication, addition, and eigenvalue calculations.
Our calculators provide high-precision calculations with up to 10 decimal places for numerical results. Exact calculations are shown when possible, especially for integer matrices, to maintain mathematical precision.
Yes, our matrix calculators fully support complex numbers. You can enter complex numbers in the form a+bi, and all operations including eigenvalues will be calculated with proper complex arithmetic.
The calculators support matrices up to 10×10 for optimal performance. Larger matrices may be supported depending on your device's capabilities, but calculations may take longer for very large matrices.
Yes, all matrix operations include detailed step-by-step solutions. You can see the intermediate steps for determinant calculations, Gaussian elimination for matrix inversion, and the complete multiplication process.
While calculations aren't automatically saved, you can copy the results and save them externally. The input and output can be exported in various formats for documentation purposes.
Yes, these calculators provide professional-grade accuracy and are suitable for engineering calculations, academic research, and professional applications. They follow standard mathematical conventions and algorithms.
Eigenvalues are calculated by finding the roots of the characteristic polynomial det(A - λI) = 0. The calculators use numerical methods for large matrices and exact methods when possible for smaller ones.