Enter transformation matrix and apply it instantly here. See inverse and determinant when square matrices. Download clean results as CSV or PDF anytime today.
Example uses a 2×2 rotation-like transform and two input vectors.
| 0 | -1 |
| 1 | 0 |
| 2 | -1 |
| 3 | 4 |
| -3 | -4 |
| 2 | -1 |
A linear transformation maps input vectors using matrix multiplication. If A is m×n and X contains k vectors as columns (n×k), then the output is:
When composition is enabled, the calculator uses C = B × A first, then Y = C × X. For square effective matrices, determinant and inverse are computed using row operations.
A linear transformation converts vectors to new vectors using fixed coefficients. In this calculator, those coefficients are stored in matrix A. Because the mapping is linear, superposition holds: A(u+v)=Au+Av and A(cu)=cAu. When you enter several input vectors, they are arranged as columns in matrix X. The output matrix Y contains transformed vectors as columns, computed in one consistent operation for speed and repeatability.
Dimension control prevents incorrect multiplication. If A is m×n, every input vector must have length n, so X is n×k. The result Y is m×k. Each entry follows yᵢⱼ = Σₜ aᵢₜ xₜⱼ, so a single wrong dimension changes the meaning, not just the size. Set k>1 to batch-test scenarios like basis vectors, sample points, or multiple parameter sets.
For square effective matrices, the determinant summarizes geometric scaling. A determinant of 2 doubles oriented area in 2D; −1 flips orientation while preserving area; 0 collapses space onto a lower dimension. In 3D, |det| scales volume; in nD, it scales hypervolume. The calculator attempts an inverse using row operations with pivoting for stability. If inversion fails or det ≈ 0, the mapping is not one‑to‑one, so some outputs cannot be uniquely traced back.
Many workflows apply transformations in sequence, such as rotate then scale, or change basis then project. Composition uses C = B×A so one combined matrix represents the full pipeline. You still provide X once, and the output becomes Y = C×X. This is numerically efficient and reduces repeated rounding when the same pipeline is reused. Practical pipelines appear in graphics, robotics coordinate frames, signal processing filters, and feature transforms for analytics.
Treat each output column as the transformed version of its matching input column. Compare norms, angles, or component ranges to validate expected scaling or rotation. When the inverse exists, multiplying A⁻¹×Y should reproduce X within rounding error. If errors grow, review input magnitudes, near-singular determinants, or extreme scaling factors. Exporting CSV or PDF helps document assumptions, inputs, and results for audits, reports, and classroom work.
k is the number of input vectors you want to transform. Each column of X is one vector of length n, and the calculator returns the matching output vectors as columns in Y.
Enable it when you need two transformations applied in order. Enter A for the first step and B for the second. The calculator forms C = B×A and applies C to every input vector.
The determinant is defined only for square matrices. If the effective matrix is not n×n, or if composition changes it to a non-square size, the determinant section shows a dash.
An available inverse means the transformation is one-to-one and reversible for the effective matrix. You can recover X from Y by multiplying A⁻¹×Y, subject to normal floating-point rounding.
Set k to the number of vectors, then enter each vector as a column in X. This batch method is faster and keeps comparisons consistent because all vectors use the same effective matrix.
It happens when A’s columns do not match X’s rows, or when B’s columns do not match A’s rows during composition. Adjust the size selectors so multiplication rules are satisfied.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.