Null Space Calculator — 2×2 Matrix Kernel
Enter the four entries of a 2×2 matrix to find its null space (kernel) — the set of all vectors mapped to zero. The calculator shows the determinant, rank, nullity and, if it exists, a basis vector that spans the null space.
Matrix A (2×2)
Non-trivial null space — a basis vector exists
- 1
Determinant
2 × 2 − 4 × 1 = 0det(A) = a × d − b × c. - 2
Rank
det = 0 → rank = 1 = 1 - 3
Nullity (rank–nullity theorem)
2 − 1 = 1
How does this calculator work?
For a 2×2 matrix, the null space is trivial ({0}) when det ≠ 0, and one-dimensional (spanned by [-b, a]ᵀ) when det = 0. The rank–nullity theorem: rank + nullity = 2. Enter matrix entries to get the determinant, rank, nullity and, when it exists, the null space basis vector.
Formula
How this is calculated
The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. For a 2×2 matrix [[a, b], [c, d]], the null space is trivial (containing only the zero vector) if and only if the determinant det(A) = ad − bc is non-zero, because the matrix then has full rank 2 and an inverse. In that case nullity = 0.
When det(A) = 0 the matrix has rank 1 (or 0 for the all-zero matrix), and by the rank–nullity theorem (rank + nullity = number of columns = 2) the nullity is 1. This means there is a one-dimensional null space spanned by a single basis vector. From the first row equation ax + by = 0 the null vector is proportional to [-b, a]ᵀ; this is normalised so the leading non-zero component equals 1, giving the reduced row-echelon form basis.
The visualisation shows the linear transformation defined by A — a unit square is mapped to a parallelogram. When det = 0 that parallelogram is degenerate (its area is zero), visually confirming that all area collapses to a line (rank 1) and the null space is non-trivial.
Frequently asked questions
The null space (or kernel) of a matrix A is the set of all vectors x satisfying Ax = 0. It is always a subspace (contains 0 and is closed under addition/scaling). For a 2×2 matrix, the null space is either just the zero vector (det ≠ 0, nullity 0) or a full line through the origin (det = 0, nullity 1).
The rank–nullity theorem states that for any m×n matrix A: rank(A) + nullity(A) = n (the number of columns). For a 2×2 matrix, rank + nullity = 2. If rank = 2 (invertible), nullity = 0. If rank = 1 (singular with non-zero entries), nullity = 1. If rank = 0 (zero matrix), nullity = 2.
For larger matrices the process is the same — reduce to row echelon form (Gaussian elimination), identify free variables and express the pivot variables in terms of them. Each free variable gives one basis vector. This calculator handles the 2×2 case analytically; for 3×3 and beyond, a step-by-step RREF approach is needed.
Also known as
TG we-Calculate Editorial Team. (2026). Null Space Calculator — 2×2 Matrix Kernel [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/null-space-calculator
TG we-Calculate Editorial Team. "Null Space Calculator — 2×2 Matrix Kernel." TG we-Calculate. 2026. https://we-calculate.com/calculator/null-space-calculator.
TG we-Calculate Editorial Team, "Null Space Calculator — 2×2 Matrix Kernel," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/null-space-calculator
@misc{wecalculate_null_space_calculator, title = {Null Space Calculator — 2×2 Matrix Kernel}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/null-space-calculator}}, year = {2026}, note = {TG we-Calculate} }
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