Linear Algebra Calculator
Enter rows with spaces or commas. Use a new line for each row. Fractions like 3/4 are accepted.
Example Data Table
| Input | Value | Use | Expected Result |
|---|---|---|---|
| Matrix A | 2 1 -1 / -3 -1 2 / -2 1 2 | Coefficient matrix | Used for solving, determinant, and inverse |
| Vector b | 8, -11, -3 | Right side vector | x = 2, y = 3, z = -1 |
| Matrix B | 1 0 2 / 0 1 1 / 2 1 0 | Second matrix | Used for A + B, A - B, and A × B |
| Scalar | 2 | Scale factor | Every entry in A doubles |
Formula Used
Matrix addition: (A + B)ij = Aij + Bij.
Matrix product: (AB)ij = Σ AikBkj.
Linear system: Ax = b is solved by row reduction of the augmented matrix [A|b].
Determinant: elimination converts A into an upper triangular matrix. The product of pivots gives det(A), adjusted for row swaps.
Inverse: Gauss-Jordan reduction changes [A|I] into [I|A-1] when A is nonsingular.
Rank: rank equals the number of pivot rows in reduced row echelon form.
Frobenius norm: ||A||F = √Σaij2.
Two by two eigenvalues: λ = (tr(A) ± √(tr(A)2 - 4det(A))) / 2.
How to Use This Calculator
- Select the operation from the dropdown menu.
- Enter Matrix A with one row per line.
- Enter Matrix B when the selected operation needs two matrices.
- Enter vector b when solving Ax = b.
- Add a scalar or matrix power when needed.
- Press Calculate to show results above the form.
- Use CSV or PDF buttons to save the output.
Linear Algebra Guide
Why this calculator helps
Linear algebra can feel complex because each result depends on structure. A small sign error can change an inverse, determinant, or system solution. This calculator keeps the process organized. It accepts matrices, vectors, scalars, and powers. It then applies standard numerical methods. You can compare outputs without changing pages.
Matrix work in one place
The tool handles common classroom and engineering tasks. You can solve Ax = b, multiply matrices, find RREF, estimate eigenvalues, and inspect rank. Addition and subtraction help with quick transformations. Scalar multiplication and powers support repeated mappings. Trace and norm checks give useful summaries for square and rectangular matrices.
Clear input format
Use one row per line. Separate values with spaces or commas. Fractions are allowed, so 1/2 becomes 0.5. Matrix B is only needed for two-matrix operations. Vector b is only needed for systems. This keeps the form clean while still allowing advanced calculations.
Understanding results
The result panel appears above the form after calculation. It shows a formatted matrix, key metrics, and a chart. The chart is useful for spotting large entries, zeros, and sign changes. The table gives dimensions, rank, scalar values, or solution components. These details make checking easier.
Study and reporting benefits
The CSV download is useful for spreadsheets. The PDF download is better for homework notes, reports, or records. The example table shows a complete system with a known answer. The formula section explains the method behind each operation. Use the calculator to learn, verify, and present linear algebra work with confidence.
Good calculation habits
Before using any result, check matrix dimensions. Many errors come from mismatched rows and columns. For multiplication, columns in A must match rows in B. For inverse and determinant tasks, A must be square. For solving systems, vector b must match the number of rows. It is also useful to compare determinant, rank, and RREF together. These values explain whether a system has one solution, no solution, or many solutions. When numbers are very large or very small, rounding can hide detail. Keep original inputs saved before exporting. Small checks make final answers safer for reports, exams, or daily projects.
FAQs
Can this calculator solve three variable systems?
Yes. Enter a square coefficient matrix in Matrix A and the constants in vector b. The calculator uses row reduction to solve Ax = b when a unique solution exists.
How should I type a matrix?
Type one row per line. Separate entries with spaces or commas. You can also use fractions like 2/3. Missing values in shorter rows are treated as zero.
Why does the inverse sometimes fail?
An inverse exists only for square nonsingular matrices. If the determinant is zero, the matrix has no inverse. The calculator will show a clear error message.
Can I multiply rectangular matrices?
Yes, when the columns of Matrix A equal the rows of Matrix B. If the dimensions do not match, multiplication is not defined.
Does it calculate eigenvalues?
For two by two matrices, it gives exact quadratic eigenvalue results. For larger square matrices, it provides a dominant eigenvalue estimate using power iteration.
What does RREF mean?
RREF means reduced row echelon form. It shows pivot positions, rank, dependencies, and simplified system structure after Gauss-Jordan elimination.
Can I download my result?
Yes. Use the CSV button for spreadsheet-friendly output. Use the PDF button for a clean printable summary of the displayed calculation.
Is this suitable for study notes?
Yes. It includes formulas, examples, charts, and downloadable outputs. It is useful for checking homework, tutorials, and quick linear algebra reviews.