Properties of Determinants

Properties of Determinants:

  1. Multiplicative Property:

    • For square matrices A and B, det(AB)=det(A)×det(B).
    • It signifies that the determinant of the product of two matrices is equal to the product of their individual determinants.
  2. Transpose Property:

    • For any square matrix A, det(A)=det(AT).
    • The determinant of a matrix and its transpose are equal.
  3. Inverse Property:

    • If A is an invertible matrix (det(A)0), then det(A1)=1det(A).
    • The determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix.
  4. Scalar Multiplication:

    • For a square matrix A of order n and scalar k, det(kA)=kn×det(A).
    • The determinant of a matrix multiplied by a scalar is equal to the determinant of the original matrix raised to the power of the matrix's order, multiplied by the scalar.
  5. Row/Column Operations:

    • Performing elementary row or column operations on a matrix changes its determinant by a scalar factor but preserves its essential value.
    • For example, interchanging rows or columns multiplies the determinant by 1.

Significance and Applications:

  1. Inversibility: A square matrix A is invertible if and only if det(A)0.

  2. Solvability of Systems: For a system of linear equations represented by a matrix AX=B, the system has a unique solution if and only if det(A)0.

  3. Geometric Interpretation: The absolute value of the determinant represents the scaling factor in transformations and encodes information about the orientation change and volume scaling in space.