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3x3 Matrix Determinant Calculator

A nonzero determinant means the three rows are independent and the matrix is invertible, and its absolute value is the volume scaling factor of the transformation. A determinant of minus 306 means the matrix is invertible and reverses orientation while scaling volumes by 306. A zero determinant would mean the equations they represent have no unique solution.

The trace, 11, is the sum of the diagonal.

3x3 Matrix Determinant Calculator: inputs

Calculated result
Determinant-306
Trace (a + e + i)11

Row 1, column 1 (a) and determinant

3x3 Matrix Determinant Calculator: Row 1, column 1 (a) and Determinant5 calculations vary the “Row 1, column 1 (a)” input while holding the other entered inputs constant.2-904-1986-3068-41410-522

Determinant from row 1, column 1 (a)

Row 1, column 1 (a)6
Row 1, column 2 (b)1
Row 1, column 3 (c)1
Row 2, column 1 (d)4
Row 2, column 2 (e)-2
Row 2, column 3 (f)5
Row 3, column 1 (g)2
Row 3, column 2 (h)8
Row 3, column 3 (i)7
Determinant-306
Trace (a + e + i)11

3x3 Matrix Determinant: formula

Determinant via cofactor expansion along the first row; trace = a + e + i

Determinant: Wolfram MathWorld

Enter the nine entries row by row: a, b, c in the first row, d, e, f in the second and g, h, i in the third. The calculator expands along the first row using cofactors: a times the minor of a, minus b times its minor, plus c times its minor. Take care with negative numbers and signs, since one wrong sign changes the result.

A matrix must be square for a determinant to exist.

3x3 Matrix Determinant: changing Row 1, column 1 (a)

Determinant: assumed Row 1, column 1 (a) scenarios.
Row 1, column 1 (a)DeterminantTrace (a + e + i)
2-907
4-1989
6 (example)-30611
8-41413
10-52215

3x3 Matrix Determinant: compare input ranges

3x3 Matrix Determinant Calculator: Determinant with each input moved on its own, every other input held at the worked-example value.
InputTested fromToDeterminant: low inputDeterminant: high inputSwing
Row 1, column 1 (a)48-198-414216 (71%)
Row 2, column 3 (f)46-260-35292 (30%)
Row 2, column 2 (e)-3-1-346-26680 (26%)
Row 1, column 3 (c)02-342-27072 (24%)
Row 3, column 3 (i)59-274-33864 (21%)
Row 3, column 2 (h)79-280-33252 (17%)
Row 1, column 2 (b)02-288-32436 (12%)
Row 3, column 1 (g)13-313-29914 (4.6%)
Row 2, column 1 (d)35-307-3052 (0.7%)

3x3 Matrix Determinant: individual input comparisons

3x3 Matrix Determinant Calculator: change one input at a time; other values remain at the stated example.
InputValueDeterminantTrace (a + e + i)
Row 1, column 1 (a)2-907
4-1989
6-30611
8-41413
10-52215
Row 1, column 2 (b)-1-27011
0-28811
1-30611
2-32411
3-34211
Row 1, column 3 (c)-1-37811
0-34211
1-30611
2-27011
3-23411
Row 2, column 1 (d)2-30811
3-30711
4-30611
5-30511
6-30411
Row 2, column 2 (e)-4-3869
-3-34610
-2-30611
-1-26612
0-22613
Row 2, column 3 (f)3-21411
4-26011
5-30611
6-35211
7-39811
Row 3, column 1 (g)0-32011
1-31311
2-30611
3-29911
4-29211
Row 3, column 2 (h)6-25411
7-28011
8-30611
9-33211
10-35811
Row 3, column 3 (i)3-2427
5-2749
7-30611
9-33813
11-37015

Determinant: Row 1, column 1 (a) against Row 1, column 2 (b)

3x3 Matrix Determinant: Determinant at combined Row 1, column 1 (a) and Row 1, column 2 (b) settings.
Row 1, column 1 (a) / Row 1, column 2 (b)-113
2-54-90-126
4-162-198-234
6-270-306-342
8-378-414-450
10-486-522-558

3x3 Matrix Determinant: values and definitions

Worked-example inputs and the results they produce for the 3x3 Matrix Determinant Calculator.
Quantity and definitionValue used
Row 1, column 1 (a)6
Row 1, column 2 (b)1
Row 1, column 3 (c)1
Row 2, column 1 (d)4
Row 2, column 2 (e)-2
Row 2, column 3 (f)5
Row 3, column 1 (g)2
Row 3, column 2 (h)8
Row 3, column 3 (i)7
Determinant-306
Trace (a + e + i)11

Common error: Dropping or mis-signing a cofactor term. The alternating positive-negative-positive sign pattern across a row is easy to lose track of by hand, and a single sign error changes the determinant's sign or magnitude without looking obviously wrong.

A zero determinant rules out a unique solution, but a singular system can still be consistent with infinitely many solutions. The right-hand side decides consistency; the determinant alone does not. Rounding and cancellation matter for nearly dependent rows.

Cofactor expansion is suitable for this size but is not a general recommendation for large numerical matrices.

Cofactor expansion reduces a 3x3 determinant to three 2x2 determinants, the same pattern that extends to any size matrix: understanding it here is what makes a 4x4 or larger determinant tractable by hand, one row or column at a time, rather than an opaque black-box operation.

Expand along a different row or column as an independent check; cofactor expansion gives the identical determinant regardless of which row or column is chosen, so two different expansions that disagree point to an arithmetic slip.

See the 2 by 2 version on the 2x2 determinant calculator, and vector products on the dot product calculator, where the cross product uses a determinant. The distance and slope page and the midpoint page work in coordinates. The rectangular prism calculator gives volumes of boxes, which a determinant generalizes.

Understand the quantities in this calculation

How do I calculate a 3x3 determinant?

Expand along the first row: multiply each entry by the determinant of the 2 by 2 matrix left after deleting its row and column, alternating plus and minus signs.

What does the sign of the determinant tell me?

A negative determinant means the transformation flips orientation, like a mirror image, and the absolute value is the volume scale factor.