Matrix Rank and Nullity Calculator
Select the matrix size and enter its values to calculate the rank and nullity.
Quick Examples
How to Use the Calculator
This online rank and nullity finder is a linear algebra tool designed to evaluate matrices from 1x1 to 8x8 in dimension (whether square or not). The mathematical engine processes your data and directly gives you the requested values, integrating all the analytical justification into a single results panel.
Configuration and data entry:
- Define the dimensions: use the initial controls to set the number of rows and columns of the matrix you want to work with.
- Enter the elements: fill in each of the grid cells. The calculator supports the use of integers, decimals, and exact fractions.
When you run the calculation, the system will display a highlighted block with the comprehensive answer to your problem. Inside this box you will find the following analysis:
- Reduced row echelon form: you will see your original matrix converted to its reduced row echelon form. This is the fundamental step to extract the subsequent properties. If you need to see the elementary row operations used to get to this point, we suggest using our reduced row echelon form calculator.
- Determining the rank: by observing the reduced matrix, the algorithm will extract the rank by counting the number of non-zero rows (those rows that have at least one non-zero element).
- Determining the nullity: finally, the tool will rely on the fundamental theorem (rank-nullity theorem or dimension theorem) to deduce the null space. You will see how this value is extracted by subtracting the rank calculated in the previous step from the total number of columns in the matrix (Nullity = n - Rank).
Solved Exercises
The following are examples of problems solved by the calculator.
Exercise 1
Calculate the rank and nullity of the following 3x3 matrix:
$$ \displaystyle \begin{bmatrix} 1 & 2 & 3 \\[0.5em] 0 & 1 & 4 \\[0.5em] 5 & 6 & 0 \end{bmatrix} $$
Result
Rank: 3
Nullity: 0
After converting the matrix to its reduced row echelon form, we get:
The rank is the number of non-zero rows in the reduced row echelon form (or row echelon form). In this case, the rank is 3.
The nullity is equal to the total number of columns in the matrix minus the rank. In this case, the matrix has 3 columns, so the nullity is 3 - 3 = 0.
If you want to see the step-by-step reduction, check out the reduced row echelon form calculator.
Exercise 2
Find the rank and nullity of the given 3x4 matrix:
$$ \displaystyle \begin{bmatrix} 1 & -2 & 1 & -1 \\[0.5em] 2 & -3 & 3 & 1 \\[0.5em] 1 & -1 & 2 & 2 \end{bmatrix} $$
Result
The rank of the matrix is 2 and its nullity is 2.
We verify this by evaluating the reduced row echelon form of the matrix:
Exercise 3
Determine the rank and nullity of the 4x3 matrix given below:
$$ \displaystyle \begin{bmatrix} 1 & 2 & 3 \\[0.5em] 2 & 3 & 4 \\[0.5em] 3 & 4 & 5 \\[0.5em] 4 & 5 & 6 \end{bmatrix} $$
Result
This matrix has a rank equal to 2 and a nullity of 1.
We can verify this by observing its corresponding reduced row echelon form matrix:
Exercise 4
Find the rank and nullity of the shown matrix:
$$ \displaystyle \begin{bmatrix} 1 & 2 & 3 \\[0.5em] 4 & 5 & 6 \\[0.5em] 7 & 8 & 9 \end{bmatrix} $$
Result
The rank is obtained as 2, while its nullity is equal to 1.
This conclusion is evident after calculating the reduced row echelon form of the matrix:
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