Determinant

Determinant is a mathematical concept used in linear algebra. It is a special number that can be calculated from a square matrix. The determinant helps us find the inverse of a matrix, tells us things about the matrix that are useful in systems of linear equations, calculus and more.

Definition

The determinant of a 2×2 matrix is defined as:

det(A) = ad − bc

where:

A = [[a, b], [c, d]]

For a 3×3 matrix, the determinant can be found by a method called "expansion by minors", or "Laplace expansion".

Properties

Determinants have some special properties, including:

• The determinant of the identity matrix is 1.
• The determinant changes sign when two rows are swapped.
• The determinant is zero if all the elements of a row or column are zero.
• The determinant is affected by row operations.

Applications

Determinants are used in a wide range of applications in mathematics and beyond, including:

• Solving systems of linear equations (Cramer's Rule)
• Calculating the inverse of a matrix
• Calculating the area or volume of a geometric shape
• In calculus, for change of variables in multiple integrals

See also

• Matrix (mathematics)
• Linear algebra
• Cramer's Rule
• Inverse matrix

References

• Area of a parallelogram as determinant
  Area of a parallelogram as determinant
• Determinant of a parallelepiped
  Determinant of a parallelepiped
• Sarrus's rule for 3x3 determinants
  Sarrus's rule for 3x3 determinants
• Jacobian determinant and distortion
  Jacobian determinant and distortion
• Determinant as a natural transformation
  Determinant as a natural transformation