How to calculate and interpret a determinant
Enter a 2 × 2 or 3 × 3 matrix
Choose the square matrix size and type every entry. A determinant is defined here only for 2 × 2 and 3 × 3 matrices, so there is one size selector rather than separate row and column controls. Zero, negative numbers, and decimals are accepted. A blank cell is not assumed to be zero; enter 0 explicitly when it is part of the matrix.
Changing the size preserves cells that remain in the upper-left overlap and leaves new cells blank. It also removes any previous determinant immediately. The same invalidation happens when a cell changes, preventing an old scalar result from being copied after the matrix has changed. Clear A empties the matrix, error, result, and copied state but retains the selected size.
Understand the formula
For A = [[a, b], [c, d]], det(A) = ad − bc. Multiply the main diagonal, multiply the other diagonal, and subtract in that order. This is not the trace: the trace is a + d, a diagonal sum, while the determinant is a signed combination of products. The page reports only the determinant so the two quantities cannot be confused.
For a 3 × 3 matrix, the calculator expands along the first row: a(ei − fh) − b(di − fg) + c(dh − eg). Each parenthesis is a 2 × 2 determinant, and the middle term has a minus sign. The numeric implementation uses this fixed formula rather than an arbitrary-size elimination engine. That keeps the assigned scope clear and makes the two examples reproducible by hand.
Interpret and check the result
A determinant of zero means the square matrix is singular: its transformation collapses area or volume, and it has no inverse. A nonzero determinant means the matrix is invertible, though this page does not calculate that inverse. The absolute value gives the area-scaling factor in two dimensions or volume-scaling factor in three dimensions. A negative sign also indicates an orientation reversal.
For a 2 × 2 result, check ad − bc directly. For a triangular 3 × 3 matrix, the determinant equals the product of the diagonal entries, which offers a quick verification case. Swapping two rows changes the determinant sign, and repeating a row makes the determinant zero. Copy result writes the displayed scalar and label for use in notes; retain the original matrix if the surrounding calculation matters.
Worked examples
Calculate a 2 × 2 determinant
For A = [[1, 2], [3, 4]], compute 1×4 − 2×3 = 4 − 6 = -2. The determinant is -2. Because it is nonzero, the matrix is nonsingular; the negative sign indicates an orientation reversal in the associated planar transformation.
Calculate a 3 × 3 determinant
For A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], expand along the first row: 1(1×0 − 4×6) − 2(0×0 − 4×5) + 3(0×6 − 1×5) = -24 + 40 − 15 = 1.
Limits and troubleshooting
Supported calculation and precision
This page accepts only 2 × 2 and 3 × 3 matrices with finite numeric entries. It does not accept 1 × 1 or larger matrices, fraction strings, variables, symbolic expressions, complex numbers, or pasted matrix notation.
The displayed determinant normally uses at most two decimal places and removes trailing zeros. A nonzero value that would otherwise round to 0 switches to compact scientific notation. Internal multiplication and addition use JavaScript numbers until final formatting. This is suitable for small numeric checks, not a proof of exact equality when rounded decimal inputs are approximations.
Errors and excluded operations
Complete every visible cell and use a period for decimal input. A number error indicates unsupported text; a range or overflow error means the fixed numeric safety bound was exceeded. Smaller scaled values can help, but remember that scaling a row also scales the determinant.
The page does not calculate an inverse, trace, rank, eigenvalues, cofactors, RREF, or symbolic expansion. It reports determinant zero normally rather than treating a singular matrix as an error. Use the separate transpose or multiplication page for those operations.
Frequently asked questions
What does determinant zero mean?
It means the matrix is singular. Its rows or columns are linearly dependent, its transformation collapses dimension, and an inverse does not exist. Zero is a valid result, not an input error.
Is the determinant the sum of the diagonal?
No. The diagonal sum is called the trace. A determinant uses signed products: ad − bc for 2 × 2 and a cofactor expansion for 3 × 3.
Can I calculate a determinant for a rectangular matrix?
No. Determinants are defined for square matrices. This focused calculator provides only 2 × 2 and 3 × 3 square inputs.
How are small decimal results displayed?
A nonzero result that is too small for the ordinary two-decimal display uses compact scientific notation instead of appearing as zero. For work requiring exact fractions or more digits, use an exact-arithmetic method and keep the original values.
Does a nonzero determinant give me the inverse?
It tells you an inverse exists, but it does not compute it. Inverse calculation and row reduction are intentionally outside this page’s assigned scope.