Properties of Determinants
Properties of Determinants:
- For square matrices and , .
- It signifies that the determinant of the product of two matrices is equal to the product of their individual determinants.
- For any square matrix , .
- The determinant of a matrix and its transpose are equal.
- If is an invertible matrix (), then .
- The determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix.
- For a square matrix of order and scalar , .
- 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.
- 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 .
Significance and Applications:
Inversibility: A square matrix is invertible if and only if .
Solvability of Systems: For a system of linear equations represented by a matrix , the system has a unique solution if and only if .
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.