3×3 Determinant Calculator
Enter the nine entries of a 3×3 matrix to compute its determinant with the rule of Sarrus. See each term and whether the matrix is invertible.
Input
Enter each entry of the 3×3 matrix.
Result
Determinant (det)
20
Sum of positive terms
29
Sum of negative terms
9
Invertible (det≠0)
Yes
Terms of the rule of Sarrus
det = aei + bfg + cdh − ceg − bdi − afh
| Term | Sign | Value |
|---|---|---|
| a·e·i | + | 27 |
| b·f·g | + | 1 |
| c·d·h | + | 1 |
| c·e·g | − | 3 |
| b·d·i | − | 3 |
| a·f·h | − | 3 |
How it works
- The determinant of a 3×3 matrix is found with the rule of Sarrus: det = aei + bfg + cdh − ceg − bdi − afh, where the entries are written as [[a,b,c],[d,e,f],[g,h,i]].
- The determinant equals the sum of the three positive terms (aei, bfg, cdh) minus the sum of the three negative terms (ceg, bdi, afh).
- If the determinant is non-zero the matrix is invertible (non-singular) and an inverse exists. A determinant of zero means the matrix is singular and has no inverse.
- Geometrically the determinant is the signed volume scaling factor of the linear transformation; a larger absolute value stretches space more.
- Because of floating-point arithmetic, tiny rounding errors are treated as zero in the displayed result.
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3×3 Determinant Calculator