Determinant Calculator

1. Choose the matrix size, 2 by 2 or 3 by 3.
2. Enter the numbers row by row, separated by commas.
3. Read the determinant, a single value.
4. See whether the matrix is invertible.

Here is what each result means:

A determinant is a single number computed from a square matrix that captures key information about it. Most importantly, it tells you whether the matrix has an inverse: a determinant of zero means no inverse exists, while any nonzero value means one does. For a 2x2 matrix with rows a, b and c, d, the determinant is a d - b c.

Geometrically, the determinant is the factor by which the matrix scales area in two dimensions or volume in three. A determinant of zero means the transformation flattens space onto a line or plane, which is why it cannot be reversed. Determinants are central to solving linear systems, finding inverses and much of linear algebra.

It applies the standard formula for the size you choose.

1. For a 2x2, multiply the diagonal and subtract the anti-diagonal.
2. For a 3x3, take each top-row entry times its 2x2 minor, with alternating signs.
3. Add those signed products for the determinant.

A nonzero determinant means an inverse exists; the underlying arithmetic uses the same multiply-and-subtract idea as the matrix multiplication calculator.

Find the determinant of the 2x2 matrix with rows 1, 2 and 3, 4.

Calculation: det = (1)(4) - (2)(3) = 4 - 6 = -2. The value is not zero, so the matrix is invertible. A 3x3 works the same way, expanding along the top row with 2x2 minors.

The sign and size of the determinant both carry meaning.

The size of the determinant is the area or volume scaling factor of the transformation the matrix represents.

The idea is the same, but the work grows with size.

Larger matrices repeat the pattern, breaking into smaller minors, which is why hand calculation gets long quickly.

Dependent Rows

If one row is a multiple or sum of others, the determinant is zero and the matrix is singular.

Row Operations

Swapping two rows flips the sign; scaling a row scales the determinant by the same factor.

Zeros on a Row or Column

A full row or column of zeros makes the determinant zero at once.

Checking Invertibility

See at a glance whether a matrix has an inverse before trying to find it.

Solving Linear Systems

A nonzero determinant means a unique solution, useful for Cramer rule.

Geometry

Read the area or volume scaling of a linear transformation.

1. Forgetting the Minus in 2x2

It is a d minus b c, not a d plus b c. The anti-diagonal is subtracted.

2. Wrong Signs in 3x3

The middle top-row term is subtracted. The signs alternate plus, minus, plus.

3. Mixing Up Rows and Columns

Enter the numbers row by row, left to right, or the matrix is wrong.

4. Wrong Count of Entries

A 2x2 needs four numbers and a 3x3 needs nine. A miscount changes everything.

5. Expecting an Inverse When Det Is Zero

A zero determinant means no inverse exists at all.

The formulas are exact; only the displayed decimals are rounded.

It is a single number computed from a square matrix. Most importantly, it tells you whether the matrix has an inverse: zero means no inverse, and any nonzero value means one exists.

Multiply the top-left and bottom-right, then subtract the product of the other two. For rows 1, 2 and 3, 4 the determinant is 1 times 4 minus 2 times 3, which is minus 2.

Expand along the top row: take each entry times the 2x2 determinant of the numbers not in its row or column, with alternating plus and minus signs, then add the results.

It means the matrix is singular: it has no inverse, its rows are linearly dependent, and a related linear system has either no solution or infinitely many.

A matrix has an inverse only if its determinant is not zero. The determinant even appears in the inverse formula, so a zero value makes the inverse undefined.

The size still gives the area or volume scaling, but a negative sign means the transformation flips orientation, like a reflection.

This tool covers 2x2 and 3x3 matrices. Larger determinants use the same idea, expanding into smaller minors, but the work grows quickly and needs a general method.

Yes. Enter the numbers row by row, left to right. Swapping entries changes the matrix and its determinant.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Determinant (Wikipedia).
• Determinant of a matrix (Maths Is Fun).
• Invertible matrix (Wikipedia).