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Free Online Matrix Calculator With Inverse and Determinant

Add, subtract, multiply, transpose, invert and reduce matrices up to eight by eight. Integer matrices give integer answers rather than figures like 2.9999999999999996, the inverse is verified by multiplying it back against the original, and a matrix that cannot be inverted is named as singular rather than returning nonsense.

Free Forever Nothing Uploaded Up To Eight By Eight The Inverse Is Checked

Free Online Matrix Calculator With Inverse and Determinant

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🔲 Matrix Calculator Up to eight by eight · integer answers stay integers · the inverse is checked

The sum runs in your browser. Nothing you type is sent anywhere.

What to work out

🔲 The matrices

One row per line, numbers separated by spaces or commas. A pasted [[1,2],[3,4]] works too. Up to eight rows and eight columns.

Matrix A

Matrix B

Multiply every entry by

Raise to the power A whole number from 0 to 12. Power zero is the identity matrix.

Try one of these

The fourth cannot be inverted and the seventh has shapes that do not fit. Both are explained rather than returning a number that means nothing.

The answer

Size of the answer—

Determinant—

Rank—

Trace—

Recent calculations, kept in this browser only.

Nothing here yet Press Log to history on the Calculate tab to keep a calculation here.

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How to Use the Matrix Calculator

Three steps, and the third is worth a look.

Choose what to work out Add, subtract, multiply, scale, transpose, determinant, inverse, row reduce or a power. The boxes you need appear and the ones you do not are hidden, so there is never an unused field to wonder about.

Type the matrices, one row per line Numbers separated by spaces or commas, or a pasted bracket form. The size is shown next to each box as you type, which catches a missing number before it becomes a confusing error.

Read the answer and the check The determinant, rank and trace come with the answer where they apply. For an inverse the check panel multiplies it back against your matrix, which is the only way to know a near singular inverse is worthless.

What to Know About Matrix Arithmetic

Shapes, singular matrices and why the digits matter.

Integer matrices give integer answers. Gaussian elimination on whole numbers leaves results like 2.9999999999999996 and minus 7.77 times ten to the minus sixteen. Those are correct to the machine and misleading on a page. Two by two and three by three inverses are computed from the adjugate instead, which keeps exact values like 0.6 exact, and elsewhere values inside the rounding tolerance are snapped while anything outside it is left alone.

A singular matrix is named, not divided by. A matrix with determinant zero has no inverse, and the formula divides by the determinant. Returning infinity or a huge number here is the classic failure. When the determinant is zero the answer is that there is no inverse, and the rank is shown so you can see how many independent rows there actually were.

Nearly singular is flagged, because it is worse. A determinant that is not zero but is tiny compared with the size of the matrix gives an inverse that exists and is numerically worthless. That case looks perfectly healthy and is the one that quietly ruins an answer, so it is called out and the check panel shows how far the product drifts from the identity.

How small counts as zero depends on the matrix. The determinant of a ten by ten matrix of numbers up to ten sits around ten to the tenth, so a value of ten to the minus nine there is genuinely zero. In a two by two of small numbers the same value is enormous. One fixed threshold is wrong in both directions, so the threshold is built from the size of your entries and the order of the matrix.

Shapes are checked before the arithmetic. Addition needs identical shapes and multiplication needs the columns of the first to match the rows of the second. When they do not, the page says which rule was broken and what the two shapes are, which is more use than the word error.

It runs in your browser with nothing uploaded. The arithmetic is in the page, so it works offline once loaded and nothing you type is sent anywhere.

Key Features & Capabilities

What it does, and what it deliberately does not.

Nine operations Add, subtract, multiply, scale, transpose, determinant, inverse, row reduce and integer powers.

The inverse is verified Multiplied back against your matrix, with the worst drift from the identity reported.

Exact where it can be Small inverses use the adjugate, so integer matrices invert to clean fractions rather than long decimals.

Rank and reduced row echelon form For any shape, not just square ones, with the pivot columns identified.

Shape errors explained Which rule was broken and what the two shapes were, rather than a bare error.

Nothing is uploaded It runs in this tab and keeps working offline.

About Matrix Arithmetic

A matrix is a rectangle of numbers, and matrix arithmetic is the language of anything with several quantities changing together: systems of equations, rotations and scalings in graphics, transition probabilities, least squares fitting. The operations themselves are simple enough to do by hand on small examples, and tedious enough beyond three by three that a calculator is the sensible tool.

Addition and subtraction need the two matrices to be the same shape. Multiplication is the one that surprises people: the columns of the first have to match the rows of the second, the answer has the rows of the first and the columns of the second, and the order matters, so A times B and B times A are usually different and are often not even the same shape. Those rules are checked here before anything is computed, and when one is broken the page says which rule and what the shapes were.

The determinant is a single number that says whether the matrix can be undone. If it is zero the matrix squashes space flat, information is lost and there is no inverse. The practical difficulty is that the determinant is computed by elimination, which leaves rounding dust, so a determinant that should be exactly zero arrives as ten to the minus sixteen instead. Treating that as non zero produces an inverse full of enormous numbers that look like an answer. The threshold for calling it zero therefore has to scale with the size of the entries and the order of the matrix, which is what is done here.

There is a second accuracy decision worth stating. For two by two and three by three matrices the inverse is computed from the adjugate rather than by elimination. That means an integer matrix inverts to clean values: you get 0.6 rather than 0.6000000000000001. It is more accurate than the general purpose routine in numpy for these small sizes, and small sizes are what most people type in.

Finally, every inverse is checked by multiplying it back against the matrix you typed and comparing with the identity. The worst drift is shown. For a well behaved matrix it is zero or very nearly so. For a nearly singular one it is large, and seeing that is the only reliable warning that the inverse on screen, however plausible it looks, should not be used.

Frequently Asked Questions

Sizes, inverses, determinants and what goes wrong.

Up to eight rows and eight columns. That covers the sizes people type by hand; beyond it the practical tool is a library rather than a web page.

Because the number of columns in the first has to equal the number of rows in the second. A three by two times a two by four works and gives a three by four. A three by two times a three by two does not, and the page says so with both shapes.

That the determinant is zero, which means the rows are not independent: one of them is a combination of the others. Such a matrix loses information, so no matrix can undo it. The rank shows how many independent rows there really were.

Because an inverse that exists on paper can still be numerically useless. If the determinant is tiny relative to the entries, small rounding differences in the input produce wildly different inverses, and the check panel shows the product drifting away from the identity.

One row per line, with numbers separated by spaces, tabs or commas. A bracketed form pasted from code, such as [[1,2],[3,4]], is also accepted. The size is shown beside the box so you can see it read what you meant.

Almost never. Matrix multiplication is not commutative, and swapping the order can even change the shape of the answer. If you want the other product, swap the two boxes.

No. Every operation runs in your browser, so nothing you type leaves the page and it works offline.