Linear Algebra
Cofactor expansion along the first row
The general rule, and the only one that scales beyond 3 by 3.
The Formula
det = a11 x (a22 x a33 - a23 x a32) - a12 x (a21 x a33 - a23 x a31) + a13 x (a21 x a32 - a22 x a31)
The general rule, and the only one that scales beyond 3 by 3.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| det | Determinant | unitless | The signed scale factor of the matrix. |
| a1j | Row entries | unitless | The entries of the row you expand along. |
| M1j | Minors | unitless | The determinant of what remains after deleting that row and column. |
What Each Symbol Means
detunitless
Determinant
The signed scale factor of the matrix.
a1junitless
Row entries
The entries of the row you expand along.
M1junitless
Minors
The determinant of what remains after deleting that row and column.
See It Worked With Real Numbers
A formula on its own does not tell you which number goes where. The how to calculate a matrix determinant page applies this equation to a concrete case, shows every intermediate step, and names the mistakes that produce a wrong answer even when the algebra is right.
Where This Formula Appears
This equation is the working core of How To Calculate A Matrix Determinant, filed under Math. Every number the guide produces comes from substituting into this one line.