Determinants Basics

  • A determinant is a special numerical value calculated only for square matrices.
  • It can be computed by applying a series of transformations and operations to the rows and/or columns of the matrix.
  • The determinant of a matrix gives us important information about the matrix, particularly in equations and systems of linear equations that the matrix represents.

Calculation of Determinants

  • The determinant of a 1x1 matrix is the single value in the matrix.
  • The determinant of a 2x2 matrix is calculated using the rule ad - bc, where a, b, c and d are the elements of the matrix.
  • For a 3x3 matrix, you can use the rule of Sarrus: repeat the first two columns of the matrix to the right of the matrix, then multiply the elements of the down-right diagonals, sum them up and subtract the sum of the products of the up-right diagonals.
  • For matrices larger than 3x3, the determinant can be calculated by the method of cofactors, minors and the expansion of a row or column.

Properties of Determinants

  • The determinant of the identity matrix is 1.
  • If we transpose a matrix (swap rows and columns), the determinant of the new matrix is the same as the determinant of the original.
  • If any row or column in a square matrix is multiplied by a scalar, then the determinant of the new matrix is the determinant of the original matrix multiplied by the scalar.
  • If a square matrix has a row or column of zeroes, its determinant is 0.
  • If two rows or columns of a matrix are identical, the determinant is 0.
  • The determinant of the product of two square matrices is the product of their determinants, i.e., Det(AB) = Det(A) * Det(B).
  • If the determinant of a matrix is 0, the matrix is said to be singular, and it has no inverse.

Applications of Determinants

  • Determinants are used to calculate the inverse of a matrix.
  • They are used to solve systems of linear equations (Cramer’s Rule).
  • They can be used to check the consistency of the system of equations.
  • They also play a crucial role in various operations in 3D geometry.