Determinants

Determinants

Basic Concepts

  • The determinant is a special numerical value calculated from a square matrix.
  • It is denoted as A or det(A) where A is the matrix.
  • The determinant of a 1x1 matrix is the lone number in the matrix.
  • The determinant of a 2x2 matrix can be computed as: (ad - bc), assuming the matrix is of the form [a,b;c,d].
  • For larger matrices (3x3 and up), determinants can be calculated using methods like Expansion by Minors, Cofactor Expansion or Row Operations.

Properties of Determinants

  • If a matrix A is square and has a row or column of zeros, then det(A) = 0.
  • Swapping two rows or two columns of A changes the sign of det(A).
  • If one multiplies all the elements of a row or a column of A by a scalar, then the determinant is multiplied by that scalar.
  • If two matrices A and B are identical apart from one row (or column), and in that row (or column) Matrix B has k times as much as Matrix A, then det(B) = k*det(A).
  • The determinant of the sum of two matrices is not usually the sum of their determinants, in other words det(A+B) != det(A) + det(B) generally.

The Determinant and Inverse Matrix

  • The inverse of a matrix A, denoted A^-1, exists only when the determinant of the matrix is not zero, i.e., det(A) ≠ 0.
  • If a matrix is singular, i.e., its determinant is zero, then it does not have an inverse.
  • The determinant of the inverse of a square matrix A is 1/det(A), i.e., det(A^-1) = 1/det(A).

Applications of Determinants

  • Determinants can be used to solve system of linear equations using Cramer’s rule.
  • The determinant helps in understanding whether a system of linear equations has a unique solution, no solution or infinitely many solutions.
  • A positive determinant suggests the matrix transformation preserves orientation, while a negative one suggests reversal of orientation.
  • It is also used to find the area of triangles, volume of parallelepipeds and to determine whether three points in a plane are collinear.

Determinant and Matrix Transformations

  • The determinant of a transformation matrix is closely related to properties of the transformation. It measures the scale factor by which the transformation changes areas or volumes.
  • If the determinant of a matrix A is 1, the matrix transformation preserves the area or volume.
  • If the determinant is negative, it indicates a reflection along with a scaling.