Linear algebra MOC

Matrix determinant

The determinant det𝐴 ∈𝕂 of a matrix 𝐴 ∈M𝑛,𝑛⁡𝕂 is a scalar quantity uniquely defined by its properties, namely: linalg

1. det𝟙 =1, where 𝟙 is the identity matrix;
2. The exchange of two rows of 𝐴 multiplies the determinant by −1;
3. Multiplying a row by a scalar multiplies the determinant by that scalar;
4. Adding any multiple of a different row to a given row does not affect the determinant.

Leibniz formula

The determinant of a matrix 𝐴 =(𝑎𝑖𝑗) ∈M𝑛,𝑛⁡𝕂 is given by linalg

det(𝑎𝑖𝑗)=∑𝜏∈S𝑛(sgn⁡𝜏)𝑛∏𝑖=1𝑎𝑖𝜏(𝑖)=∑𝜏∈S𝑛(sgn⁡𝜏)𝑛∏𝑖=1𝑎𝜏(𝑖)𝑖

which is known as the Leibniz formula for the determinant.

Proof

proof

See also

• Vandermonde determinant

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