Coefficient matrix determinant
- Input
- det([[1, 2], [3, -1]])
- Expected output
- -7
Nonzero, so Cramer's rule applies and the system has a unique solution.
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cramer's rule
Cramer's rule, named after the Swiss mathematician Gabriel Cramer (who published it in 1750), solves a square linear system directly from determinants, without going through elimination rows. This page opens a sample 2x2 system with a nonzero determinant.
Nonzero, so Cramer's rule applies and the system has a unique solution.
Each unknown is the determinant of the matrix with that column substituted, divided by the original determinant.
One equation per line, with an equals sign separating the two sides, for example "2x + 3y = 8" or "x - y = -1". The calculator accepts an implicit coefficient ("x" means the same as "1x"), a negative sign, unknowns with any name, a variable appearing on both sides of the equals sign ("2x + 1 = x + 4"), and a variable that simply doesn't appear in one of the equations, it is treated as a zero coefficient automatically.
No, it requires a square system. When the number of equations and unknowns differ, this calculator states that explicitly and still solves it via Gaussian elimination, shown alongside.
It means the rule cannot be used for that system: it is either inconsistent (no solution) or consistent and dependent (infinitely many solutions). The calculator identifies which of the two cases it is through Gaussian elimination.
calculadoraSistemaEquacoes.classificationDesc.determined
Matrix representation
[ 1 2 ] [ 3 -1 ]
[ x ] [ y ]
[ 5 ] [ 1 ]
[ 1 2 | 5 ] [ 3 -1 | 1 ]
[ 1 0 | 1 ] [ 0 1 | 2 ]
[ 3 -1 | 1 ] [ 1 2 | 5 ]
[ 1 -1/3 | 1/3 ] [ 1 2 | 5 ]
[ 1 -1/3 | 1/3 ] [ 0 7/3 | 14/3 ]
[ 1 -1/3 | 1/3 ] [ 0 1 | 2 ]
[ 1 0 | 1 ] [ 0 1 | 2 ]
Substitution check
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