Determinant of Orthogonal Matrix
An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors, meaning that the dot product of any two distinct rows or columns equals zero, and the dot product of each row or column with itself equals one. Mathematically, if A is an orthogonal matrix, then AT⋅A=I, where AT denotes the transpose of A and I represents the identity matrix.
The determinant of an orthogonal matrix has a special property:
det (A) = ±1
The determinant of an orthogonal matrix is either +1+1 or −1−1. This property arises from the fact that the determinant represents the scaling factor of the matrix transformation. Since orthogonal transformations preserve lengths, the determinant must be either positive (for preserving orientation) or negative (for reversing orientation).
The determinant of an orthogonal matrix being +1+1 implies that the transformation preserves orientation, while a determinant of −1−1 indicates a transformation that reverses orientation.
Determinant of a Matrix with Solved Examples
Determinant of a Matrix is defined as the function that gives the unique output (real number) for every input value of the matrix. Determinant of the matrix is considered the scaling factor that is used for the transformation of a matrix. It is useful for finding the solution of a system of linear equations, the inverse of the square matrix, and others. The determinant of only square matrices exists.
Table of Content
- Determinant of Matrix Calculator
- Definition of Determinant of Matrix
- Determinant of a 1×1 Matrix
- Determinant of 2×2 Matrix
- Determinant of a 3×3 Matrix
- Determinant of 4×4 Matrix
- Determinant of Identity Matrix
- Determinant of Symmetric Matrix
- Determinant of Skew-Symmetric Matrix
- Determinant of Inverse Matrix
- Determinant of Orthogonal Matrix
- Physical Significance of Determinant
- Laplace Formula for Determinant
- Properties of Determinants of Matrix