Enter Matrix
Example Data Table
| Matrix size | Rows entered | Expected rank | Expected nullity | Kernel note |
|---|---|---|---|---|
| 3 × 4 | [1, 2, -1, 0], [2, 4, -2, 0], [0, 1, 3, 1] | 2 | 2 | Two basis vectors should appear. |
| 2 × 3 | [1, 1, 1], [2, 3, 5] | 2 | 1 | One free variable creates one basis vector. |
| 3 × 3 | [1, 0, 0], [0, 1, 0], [0, 0, 1] | 3 | 0 | The kernel is only the zero vector. |
Formula Used
The calculator solves the homogeneous equation A x = 0. It row reduces A to RREF(A). Pivot variables are written in terms of free variables.
Rank is the number of pivot columns. Nullity is calculated as n - rank(A), where n is the number of matrix columns.
For each free variable, one basis vector is formed. That variable is set to one. Other free variables are set to zero.
How to Use This Calculator
- Choose the number of matrix rows and columns.
- Enter every matrix value in the generated boxes.
- Use fractions, decimals, or integers.
- Set precision and tolerance if your data needs special handling.
- Press the calculate button to show the kernel above the form.
- Download the CSV or PDF file for records.
Kernel of Matrix Calculator Guide
What the kernel means
The kernel of a matrix is the set of all vectors that become zero after multiplication by the matrix. It is also called the null space. For a matrix A, the kernel contains every vector x that satisfies Ax = 0. This idea connects matrices, linear systems, transformations, rank, and dimension.
Why the result matters
A kernel describes hidden freedom in a homogeneous system. When the kernel has only the zero vector, the columns are independent. When the kernel has nonzero vectors, there are dependent columns. Each basis vector shows one independent direction inside the solution set. The nullity tells how many such directions exist.
How row reduction helps
This calculator finds the kernel by reducing the matrix to reduced row echelon form. Pivot columns control leading variables. Non pivot columns become free variables. Each free variable is assigned a parameter. The calculator then builds basis vectors by setting one parameter to one and all other parameters to zero.
Reading the output
The rank counts pivot columns. The nullity equals the number of columns minus the rank. A zero nullity means the kernel is trivial. A positive nullity means infinitely many solutions exist. The displayed basis can be combined with any scalar weights to form every vector in the kernel.
Practical use
Students can use this tool to check homework, test transformations, and study column dependence. Engineers and data analysts can inspect systems with redundant equations. The calculator accepts decimals and fractions, so exact classroom examples and measured data both work.
Good input habits
Enter each row carefully. Keep the same number of entries in every row. Fractions like 3/4 are accepted. Very small decimal values are treated with tolerance during pivot checks. After calculation, review the RREF first, then read pivot and free columns. Export the result when you need a record.
Advanced checks
Use the decimal precision control to round displayed values. Choose a tolerance that fits your data. Larger tolerance can hide tiny noise. Smaller tolerance preserves delicate pivots. The verification section multiplies A by each basis vector. A correct kernel basis produces zero entries, except for rounding limits. This check helps catch input mistakes before saving results later.
FAQs
What is the kernel of a matrix?
It is the set of all vectors x that satisfy A x = 0. It is also called the null space of the matrix.
What does nullity mean?
Nullity is the dimension of the kernel. It equals the number of columns minus the rank of the matrix.
Can this calculator handle non square matrices?
Yes. The kernel is defined for rectangular and square matrices. The vector length always matches the number of columns.
What does a trivial kernel mean?
A trivial kernel means only the zero vector solves A x = 0. This happens when every column is a pivot column.
Why are free variables important?
Free variables create the independent directions of the kernel. Each free variable usually gives one basis vector.
Can I enter fractions?
Yes. You can enter values such as 1/2, -3/4, or 5/6. The calculator converts them during reduction.
What tolerance should I use?
Use a small tolerance for exact work. Use a larger tolerance when decimal data contains measurement noise or rounding errors.
How is the basis verified?
The calculator multiplies the original matrix by each basis vector. The product should be the zero vector within rounding limits.