Determinant Calculator

Calculate the scalar determinant of a square matrix and track every cofactor sign or row-operation effect.

Drag & drop or upload an image or PDF

Enter a square matrix, such as det([[2,1],[5,3]])

Enter rows in one line, such as [[1, 2], [3, 4]]. State the operation when the page accepts more than one.

Try an example

A determinant is one number from a square matrix

The determinant records signed scale change and gives a quick test for invertibility. A zero determinant means the matrix is singular, so its columns are linearly dependent and an ordinary inverse does not exist.

Small matrices are often easiest by a direct formula or cofactor expansion. Row reduction scales better, but every row swap and row scaling must be recorded because those operations change the determinant.

How to use the determinant calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Determinant of a 2 × 2 matrix

Multiply along the main diagonal, subtract the product along the other diagonal, and return the scalar value.

det [a c b d] = a d - b c

A reliable way to work through it

Confirm the matrix is square

A determinant is defined here only when the matrix has the same number of rows and columns.

Choose an efficient expansion

Use a row or column with zeros for cofactors, or use elimination for a larger dense matrix.

Track row-operation effects

A row swap flips the sign, row scaling changes the determinant, and row replacement leaves it unchanged.

Worked example

Find the determinant of A

A = 2 - 1 0 134021

Expand along the first row because it contains a zero.

det (A) = 23421 - 1 - 1 0 21

Apply the alternating cofactor signs.

det (A) = 2 (3 - 8) - ((- 1) - 0) = - 10 + 1 = - 9

Evaluate each 2 × 2 minor and simplify.

The determinant is -9, so the matrix is invertible.

Common mistakes to check

Using a non-square matrix

The determinant in standard linear algebra is defined only for square matrices.

Forgetting cofactor signs

The signs alternate in a checkerboard pattern beginning with positive in the top-left position.

Ignoring a row swap

Every row exchange multiplies the determinant by -1.

Matrix CalculatorPerform matrix operations with dimension checks, exact entries, and each row or column calculation shown.Inverse Matrix CalculatorFind A⁻¹ with exact row operations, detect a singular matrix, and verify the result with the identity.Eigenvalue CalculatorFind the characteristic polynomial, eigenvalues, and eigenspaces of a square matrix.Systems of Equations CalculatorSolve two or more equations together with substitution, elimination, or row operations.

Questions students ask

What does a determinant tell me?

It measures signed scale change and shows whether a square matrix is invertible.

What does determinant zero mean?

The matrix is singular, its columns are linearly dependent, and it has no ordinary inverse.

Does swapping two rows change the determinant?

Yes. One row swap multiplies the determinant by -1.

Which method is best for a large matrix?

Elimination is usually more efficient. Cofactor expansion is useful when a row or column has many zeros.