[A] 3×3
×
Free online matrix calculator — compute RREF, determinant, inverse, eigenvalues, and matrix multiplication for matrices up to 20×20. Works in your browser with no download or sign-up. All operations are free with no size limits.
This is a free browser-based matrix calculator that handles everything a TI-84's MATRX menu does — RREF, inverse, determinant, eigenvalues — with matrices up to 20×20 and no paywalls. If you are taking linear algebra or any course that involves systems of equations, this calculator saves you from pages of arithmetic. Unlike tools that stop at 3×3, this one goes to 20×20, and the RREF button is never locked behind a paywall. Just enter your numbers and get the result.
Students use this tool to check homework, verify exam review problems, and work through linear regression computations without getting lost in row operations. The interface matches the matrix editor on a physical TI-84: pick your matrix letter (A through J), set the dimensions, fill in the cells, and choose an operation. If you have used the MATRX button on a TI-84 Plus CE, you already know how this works.
If you have never worked with matrices before, here is the quickest way to get started. It takes about a minute.
Matrix operations have specific jobs. Here is what each one is actually for, in plain language:
RREF (Reduced Row Echelon Form) is how you solve systems of linear equations. You write your system as an augmented matrix — coefficients on the left, constants on the right — and run RREF. The calculator performs Gaussian-Jordan elimination and returns a matrix where each row has a leading 1 with zeros everywhere else. The solution to your system is in the last column. For example, a 3×4 augmented matrix after RREF tells you the values of x, y, and z in one step. This is the same as the rref() command on a TI-84.
Matrix Inverse (A⁻¹) is like division for matrices. If you have Ax = b and A is a square matrix, multiplying both sides by A⁻¹ gives you x = A⁻¹b. The inverse exists only if the determinant is not zero — that is why checking the determinant first matters. This calculator shows the determinant alongside the inverse so you can verify the matrix is invertible before relying on the result.
Determinant tells you whether a matrix is invertible (nonzero determinant) or singular (zero determinant). Geometrically, the determinant of a 2×2 matrix equals the area of the parallelogram formed by its column vectors. For a 3×3, it equals the volume of the parallelepiped. If the determinant is zero, the columns are linearly dependent — they lie on the same line or plane and do not span the full space.
Eigenvalues are the special numbers λ where A − λI becomes singular. They appear in differential equations, vibration analysis, Google's PageRank algorithm, and principal component analysis in statistics. This calculator returns both real and complex eigenvalues with their multiplicities, matching what the TI-84's eigVl() command produces.
Here is a concrete example you can follow along with right now. Say you have this system of equations:
Step 1: Write the augmented matrix. Coefficients on the left, constants on the right:
Step 2: Create a 2×3 matrix in the calculator and enter those numbers.
Step 3: Click RREF. The output will be:
Step 4: Read the answer: x = 1, y = 2. Verify by plugging back in: 2(1) + 3(2) = 2 + 6 = 8 ✓ and 4(1) − 2 = 2 ✓.
This same process works for 3×3 and larger systems. For a full walkthrough of the matrix menu on a handheld, see our TI-84 graphing calculator guide.
Yes, completely free. Every operation — RREF, inverse, determinant, eigenvalues, and multiplication — works without signup, download, or payment. Most free matrix tools cap you at 3×3 or hide RREF behind a paywall. This one handles up to 20×20 with no restrictions.
Yes. For square matrices up to 20×20, the calculator returns both real and complex eigenvalues with their algebraic multiplicities. For symmetric matrices, you also get the eigenvectors. The results follow the same conventions as the TI-84 Plus CE's built-in eigVl() and eigVc() commands.
Up to 20 rows by 20 columns. That covers every problem in a standard undergraduate linear algebra course — including augmented matrices like 3×4 for a 3-variable system or 5×6 for a 5-variable system. Most competing free tools stop at 3×3 or 5×5.
Yes. Enter the coefficient matrix and constant vector as an augmented matrix — for 3 equations with 3 unknowns, create a 3×4 matrix where the first 3 columns are coefficients and the last column is the constants. Run RREF and read the solution from the last column. This is exactly the same workflow as MATRX → MATH → rref() on a TI-84 Plus CE.
Yes. The matrix editor is fully responsive. On a phone screen, cells are large enough to tap accurately, and the matrix scrolls horizontally if it is wider than the display. All operations — RREF, inverse, determinant, eigenvalues — work identically on mobile and desktop browsers.
Type fractions using the division key — for example, 1/2 enters 0.5. Decimal numbers work directly, and results are shown with full floating-point precision. For a guide to data entry and working with numbers on the TI-84, see the linear regression tutorial.
RREF (Reduced Row Echelon Form) goes one step further than regular row echelon form. In RREF, every leading 1 is the only non-zero entry in its column, and all leading 1s equal 1. Regular row echelon form only requires zeros below each leading entry. RREF gives you the solution directly — the last column IS your answer. The TI-84's rref() command and this calculator both use RREF, which is what most homework problems expect.
Also useful: our step-by-step linear regression tutorial shows the matrix math behind least-squares fitting. The statistics calculator handles 1-Var Stats and hypothesis testing. Need to graph equations? The graphing calculator plots up to six functions at once. For a deeper dive into linear algebra concepts, Georgia Tech's interactive linear algebra textbook is an excellent free resource. See also AMS education resources for curriculum-aligned materials.
Page last updated: August 4, 2026
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