formula.expert

Determinant of a 3×3 Matrix

D=a11(a22a33−a23a32)−a12(a21a33−a23a31)+a13(a21a32−a22a31)D = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})

Worked example: [[6,1,1],[4,−2,5],[2,8,7]] → determinant −306 — press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value. Try different units for next level excitement!

Learning zone

Determinant of a 3×3 Matrix explained

D =a11a12a13a21a22a23a31a32a33

Cofactor expansion walks along the top row: take each entry, multiply it by the determinant of the 2×2 block left when you delete its row and column, and alternate the signs plus, minus, plus. For the standard textbook matrix with rows (6, 1, 1), (4, −2, 5) and (2, 8, 7): 6(−14 − 40) − 1(28 − 10) + 1(32 + 4) = −324 − 18 + 36 = −306. The answer is the signed volume of the parallelepiped spanned by the three rows, so zero means the rows are coplanar and the matrix squashes space flat — no inverse, and a linear system with no unique solution.

Determinants predate matrices by nearly two centuries. Seki Takakazu described them in Japan in 1683, and Leibniz set them out in a letter to l'Hôpital the same year, both while trying to decide when systems of equations are solvable; the word matrix itself waited until Sylvester coined it in 1850. Pierre Frédéric Sarrus gave the 3×3 diagonal shortcut in 1833 — and there lies the trap, because Sarrus's rule works for 3×3 and for nothing larger. The alternating signs are the other habitual slip: the middle term is subtracted, not added. Because D is linear in every entry, any single missing entry can be recovered — the rows (1, 2, 3), (4, ?, 6), (7, 8, 9) give a determinant of zero only when the missing entry is 5.

Determinant of a 3×3 Matrix formula

D=a11(a22a33−a23a32)−a12(a21a33−a23a31)+a13(a21a32−a22a31)D = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})
Where
  • DD= Determinant
  • a11a_{11}= Row 1, column 1
  • a12a_{12}= Row 1, column 2
  • a13a_{13}= Row 1, column 3
  • a21a_{21}= Row 2, column 1
  • a22a_{22}= Row 2, column 2
  • a23a_{23}= Row 2, column 3
  • a31a_{31}= Row 3, column 1
  • a32a_{32}= Row 3, column 2
  • a33a_{33}= Row 3, column 3

Missing one of these? Work it out first, then come back