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Null Space (Nullity) Calculator — 3×3 Matrix

Calculate the null space (kernel) and nullity of a 3×3 matrix. Enter matrix entries to instantly find all solutions to Ax = 0 using the rank-nullity theorem.

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Inputs

Row 1, Column 1

Row 1, Column 2

Row 1, Column 3

Row 2, Column 1

Row 2, Column 2

Row 2, Column 3

Row 3, Column 1

Row 3, Column 2

Row 3, Column 3

Nullity (Dimension of Null Space)

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Nullity (Dimension of Null Space)—

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Understanding the Null Space of a 3×3 Matrix

The null space of a matrix A, also called the kernel, is the set of all vectors x in R³ that satisfy the homogeneous equation Ax = 0. For a 3×3 matrix A, the null space is formally defined as: Nul(A) = {x ∈ R³ : Ax = 0}. This subspace captures every solution to the homogeneous linear system and plays a central role in linear algebra, differential equations, control theory, and data science.

The Rank-Nullity Theorem

The foundation of this calculator is the Rank-Nullity Theorem, which states:

nullity(A) = n − rank(A)

For a 3×3 matrix, n = 3 (the number of columns), so nullity equals 3 minus the rank of A. According to Gilbert Strang's MIT Linear Algebra text (Section 3.2), the rank counts pivot columns in reduced row echelon form (RREF), while the nullity counts free variables — columns without a pivot. This single equation connects two of the most important structural properties of any linear map.

How the Calculator Computes the Null Space

Step 1: Form the Augmented Matrix

Given the 3×3 matrix A with entries a11 through a33, construct the augmented matrix [A | 0] by appending a zero column. This represents the homogeneous system Ax = 0 in matrix form, setting up the system for elimination.

Step 2: Apply Row Reduction to RREF

Perform Gaussian elimination with back-substitution to convert [A | 0] into reduced row echelon form. Each elementary row operation — swapping rows, scaling a row, or adding a multiple of one row to another — preserves the solution set exactly. As demonstrated in Khan Academy's null space series, identifying pivot positions in RREF simultaneously determines both the rank and the free variables.

Step 3: Identify Free Variables and Rank

Columns without a leading 1 (pivot) in the RREF correspond to free variables. For a 3×3 matrix, four distinct cases arise:

Rank 3, Nullity 0: The matrix is invertible. The only solution to Ax = 0 is the trivial solution x = 0. The null space is {0}.
Rank 2, Nullity 1: One free variable exists. The null space is a line through the origin in R³, spanned by one basis vector.
Rank 1, Nullity 2: Two free variables exist. The null space is a plane through the origin in R³, spanned by two linearly independent basis vectors.
Rank 0, Nullity 3: A is the zero matrix. Every vector in R³ satisfies Ax = 0; the null space is all of R³.

Step 4: Express the General Solution

Write each pivot variable in terms of the free variables using back-substitution. Each free variable generates exactly one basis vector for the null space. The general solution is an arbitrary linear combination of these basis vectors, parameterized by the free variable values.

Worked Example

Let A = [[1, 2, 3], [2, 4, 6], [3, 6, 9]]. Rows 2 and 3 are scalar multiples of row 1, so RREF has a single pivot in column 1, giving rank = 1. Free variables are x2 and x3. Setting (x2, x3) = (1, 0) gives basis vector v1 = (−2, 1, 0); setting (x2, x3) = (0, 1) gives v2 = (−3, 0, 1). The null space is Span{(−2, 1, 0), (−3, 0, 1)}, confirming nullity = 2 = 3 − 1. Substituting either vector back into A confirms Av1 = 0 and Av2 = 0.

Applications and Significance

Null space analysis appears throughout science and engineering. In control systems, null space vectors identify redundant actuator configurations. In machine learning and signal processing, null space methods underpin dimensionality reduction and image compression. Georgia Tech's Interactive Linear Algebra describes how the Rank-Nullity Theorem anchors the Fundamental Theorem of Linear Algebra, connecting all four fundamental subspaces. A nullity of 0 guarantees that Ax = b has at most one solution; a positive nullity means infinitely many solutions exist whenever any solution exists at all.

Variable Reference

a11–a33: The nine scalar entries of the 3×3 matrix A, indexed by row then column.
rank(A): Number of linearly independent rows or columns; equals the number of pivot positions in RREF.
nullity(A): Dimension of the null space; equals 3 − rank(A) for any 3×3 matrix.
Nul(A): The null space itself — the complete solution set of Ax = 0, always a subspace of R³.

Reference

Frequently asked questions

What is the null space of a matrix?

The null space of a matrix A is the set of all vectors x such that Ax = 0. It is also called the kernel of A. For a 3×3 matrix, the null space is always a subspace of R³, meaning it contains the zero vector and is closed under vector addition and scalar multiplication. Its dimension is called the nullity of A and can range from 0 to 3.

How do you calculate the null space of a 3×3 matrix step by step?

Form the augmented matrix [A | 0] and reduce it to reduced row echelon form (RREF) using Gaussian elimination. Identify columns without pivot positions — these correspond to free variables. For each free variable, set it equal to 1 while all other free variables equal 0, then solve for the pivot variables. Each resulting solution vector is a basis vector of the null space. The collection of all such basis vectors spans the complete null space.

What does nullity tell you about a linear system?

Nullity measures the dimension of the null space and equals the number of free variables in the homogeneous system Ax = 0. A nullity of 0 means only the trivial solution x = 0 exists, so Ax = b has at most one solution. A nullity of 1 or more indicates infinitely many solutions exist for Ax = 0. When Ax = b has a particular solution, adding any null space vector to it yields another valid solution, producing an infinite solution family.

What is the Rank-Nullity Theorem and why does it matter?

The Rank-Nullity Theorem states that for any m×n matrix A, rank(A) + nullity(A) = n, where n is the number of columns. For a 3×3 matrix, this means rank + nullity always equals 3. The theorem matters because it quantifies the trade-off between the information a linear transformation preserves (rank) and the information it collapses to zero (nullity). It also proves definitively that a square matrix is invertible if and only if its nullity equals 0.

What does a nullity of 0 mean for a 3×3 matrix?

A nullity of 0 means the null space contains only the zero vector, so Ax = 0 has no non-trivial solutions. This is equivalent to the matrix having rank 3, meaning all three columns are linearly independent and the matrix is invertible with a non-zero determinant. In practical terms, a nullity of 0 guarantees that the linear transformation represented by A is injective (one-to-one), and the system Ax = b has exactly one unique solution for every right-hand side vector b in R³.

What is the difference between null space and column space?

The null space of A is the set of all input vectors mapped to zero: Nul(A) = {x : Ax = 0}, a subspace of R³ with dimension equal to the nullity. The column space (range) of A is the set of all possible output vectors Ax, a subspace of R³ with dimension equal to the rank. For a 3×3 matrix, rank + nullity = 3 by the Rank-Nullity Theorem. The column space determines which right-hand sides b make Ax = b solvable; the null space determines how many solutions exist once solvability is confirmed.