Matrix Determinant Calculator

What this calculator does

The determinant is a single number squeezed out of a square matrix, and for such a small thing it says a great deal: whether the matrix can be inverted, and by how much it stretches or shrinks space. This computes it for 2 by 2, 3 by 3 and 4 by 4 matrices.

Pick the size, enter the numbers, and it returns the determinant. It runs right here in the browser.

Using the calculator

1. Choose the matrix size: 2x2, 3x3 or 4x4.
2. Enter the numbers row by row.
3. Press Calculate.

It returns the determinant, a single number that can be positive, negative or zero. Reset clears the boxes.

What a determinant is

A determinant is one number that summarises a whole square matrix. Only square matrices have one, those with equal rows and columns, because the idea does not make sense for a lopsided grid. It is written either as det(A) or by wrapping the matrix in straight vertical bars, like |A|.

Small as it is, it is one of the most informative quantities in linear algebra. Whether it is zero or not decides whether the matrix has an inverse, and its size and sign carry a clear geometric meaning, which is coming up shortly.

The 2 by 2 formula

For the smallest case, a 2 by 2 matrix [[a, b], [c, d]], the determinant is beautifully simple:

det = ad − bc

Multiply the two numbers on the main diagonal (top-left and bottom-right), then subtract the product of the other diagonal (top-right and bottom-left). That is the whole calculation for a 2 by 2.

Bigger matrices: breaking it down

For a 3 by 3 or 4 by 4, the determinant is built out of smaller ones. You take each number along the top row, multiply it by the determinant of the little matrix left when you cross out that number's row and column, and add these up with alternating plus and minus signs. So a 3 by 3 determinant becomes three 2 by 2 determinants, and a 4 by 4 becomes four 3 by 3s. It grows quickly by hand, which is exactly why a tool earns its keep here, but the principle is always the same: a big determinant is assembled from smaller ones.

What the number actually tells you

Here is the part that makes the determinant click. Think of a matrix as a transformation that stretches and skews space. The determinant is the factor by which it scales area (in 2D) or volume (in 3D). A determinant of 2 means the transformation doubles areas; a determinant of one half means it halves them. A negative determinant means it also flips orientation, turning shapes into their mirror image.

And the most important case of all is a determinant of zero. That means the transformation squashes space completely flat, collapsing it onto a line or a point, so area or volume is crushed to nothing. Once space has been flattened like that, there is no way to un-flatten it, which is precisely why a matrix with a zero determinant has no inverse. The single number tells you, at a glance, whether the matrix can be reversed.

A worked example

Take the 2 by 2 matrix [[3, 8], [4, 6]].

det = (3 × 6) − (8 × 4) = 18 − 32 = -14. It is not zero, so this matrix can be inverted. The negative sign tells us the transformation flips orientation, and the size, 14, tells us it magnifies areas by a factor of 14.

Questions people ask

What is the determinant of a matrix?

A single number calculated from a square matrix that reveals whether it can be inverted and how it scales area or volume.

How do you find a 2 by 2 determinant?

For [[a, b], [c, d]], it is ad minus bc: multiply the main diagonal and subtract the product of the other diagonal.

What does a determinant of zero mean?

The matrix collapses space flat and cannot be reversed, so it has no inverse. Such a matrix is called singular.

What does the determinant represent geometrically?

The factor by which the matrix scales area or volume. A negative value also flips orientation, and zero flattens space entirely.

Why only square matrices?

The determinant measures how a matrix transforms space into itself, which only makes sense when the number of rows equals the number of columns.

References

A note on where this comes from. The determinant is a scalar computed from a square matrix that is non-zero exactly when the matrix is invertible, and whose absolute value gives the factor by which the matrix scales area or volume. Determinants are actually older than matrices, studied in the 18th and early 19th centuries by mathematicians including Leibniz, Cauchy and Jacobi. For further reading, see Determinant.

1. The determinant, det(A), zero exactly when a matrix is singular, with its magnitude giving the area or volume scaling factor.
2. The theory of determinants, developed by Cauchy and Jacobi among others, predating the formal theory of matrices.