Matrix Rank & Nullity

Rank, nullity, and reduced row echelon form

About this calculator

The Matrix Rank & Nullity Calculator row-reduces any matrix up to 6×6 to reduced row echelon form (RREF) by Gauss-Jordan elimination and reads off the rank (the number of pivots), the nullity (columns minus rank, by the rank-nullity theorem), and the pivot columns. RREF entries are reconstructed as exact fractions rather than fuzzy decimals. It also classifies the matrix: full rank, full row or column rank, the zero matrix, and — for square matrices — whether it is invertible.

How to use the Matrix Rank & Nullity calculator

  1. Enter the values for your problem into the input fields.
  2. Read the result — it updates instantly as you type.
  3. Check the formula and the visual explanation to follow how the answer was found.
  4. Copy the page URL to share the exact calculation.

Common examples

  • [[1,2,3],[4,5,6],[7,8,9]] → rank 2, nullity 1: the rows are dependent, so it isn't invertible
  • 2×3 matrix [[1,0,2],[0,1,3]] → rank 2 (full row rank), nullity 1
  • diag(2, 3, 4) → rank 3, full rank, invertible
  • The 2×2 zero matrix → rank 0, nullity 2

Frequently asked questions

How does the Matrix Rank & Nullity calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Matrix Rank & Nullity calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.