Determinant calculator with steps

The determinant is a single number worked out from a square matrix. For a 2 by 2 matrix [a, b; c, d], with a semicolon between the rows, it is ad − bc. It tells you whether the matrix has an inverse (only if the determinant isn't 0) and how much it scales areas or volumes. Our determinant calculator is part of the Matrix Calculator: choose Determinant, type a square matrix up to 5 by 5, and it gives the exact value with the working.

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How to find a determinant in the calculator

1. Open the Matrix Calculator and choose the Determinant chip. It opens on Eigenvalues, so this step matters.
2. Set the size of A with the size buttons, from 1 to 5. Determinants only exist for square matrices.
3. Type the entries. Fractions (type a slash) and decimals work, and the answer stays exact.
4. Read det A at the top, with a decimal beside it if it is a fraction.
5. Follow the working. A 2 by 2 uses ad − bc, a 3 by 3 is expanded along the top row, and a 4 by 4 or 5 by 5 is row reduced to a triangle.
6. For a 4 by 4 straight away, press Determinant 4×4 in the Examples section. Random problem in the Practice section asks for 2 by 2 and 3 by 3 determinants.

Determinant of a 2x2 matrix

Multiply down the main diagonal, then subtract the product of the other diagonal. For [7, 3; 2, 5], writing each matrix row by row, det = 7 × 5 − 3 × 2 = 35 − 6 = 29. With a fraction, [1/2, 3; −4, 2] gives 1/2 × 2 − 3 × (−4) = 1 + 12 = 13.

Determinant of a 3x3 matrix

Expand along the top row. Each entry multiplies the 2 by 2 determinant left when you cover its row and column (its minor), and the signs go plus, minus, plus:

Take this matrix:

Adding them, det A = −6 + 0 + 22 = 16. A zero in the top row saves work, because its term is 0 whatever the minor is.

4x4 and bigger: row reduction

Cofactor expansion of a 4 by 4 needs four 3 by 3 determinants, and it gets much worse after that. Instead, the calculator uses row operations to make the matrix triangular, with zeros below the diagonal. Adding some multiple of a row onto a different row leaves the determinant alone, and swapping two rows flips its sign. Once it is triangular, the determinant is the product of the diagonal entries. The calculator's 4 by 4 example, with rows 1, 2, 0, 1 / 3, 1, 4, 0 / 2, 0, 1, 5 / 1, 1, 1, 1, comes out as 12.

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What the determinant tells you

• Zero means no inverse. [2, 6; 1, 3] has det = 6 − 6 = 0: its second row is half its first, so the matrix is singular. The inverse matrix calculator guide shows what happens otherwise.
• Its size is the area scale factor. [3, 1; 0, 2] has det = 6, so it turns a unit square into a shape of area 6. The calculator's 2×2 transformation mode draws this.
• A negative determinant means a flip. The transformation turns the plane over, like a reflection.
• For a system of equations, a determinant of 0 means there is no unique solution.

Properties of determinants

Common mistakes

• Adding instead of subtracting in ad − bc.
• Losing the minus on the middle term of a 3 by 3 expansion. The signs alternate: plus, minus, plus.
• Sign slips with negative entries. Write each product in brackets before you simplify.
• Trying it on a non-square matrix. Only square matrices have determinants.
• Using the diagonal shortcut on a 4 by 4. The "diagonals" trick for 3 by 3 matrices doesn't work for bigger ones.

Read next

Maths · AlgebraMatrix calculator: multiply, inverse, determinant, RREFAdd, multiply, invert and row reduce matrices up to 5 by 5 with exact fractions, plus determinants, eigenvalues and worked examples to check.