Determinant of Orthogonal Matrix
Determinant of any Orthogonal Matrix is either +1 or -1. Here, let’s demonstrate the same. Imagine a matrix A that is orthogonal.
For any orthogonal matrix A, we know
A · AT = I
Taking determinants on both sides,
det(A · AT) = det(I)
⇒ det(A) · det(AT) = 1
As, determinant of identity matrix is 1 and det(A) = det(AT)
Thus, det(A) · det(A) = 1
⇒ [det(A)]2 = 1
⇒ det(A) = ±1
Orthogonal Matrix
A Matrix is an Orthogonal Matrix when the product of a matrix and its transpose gives an identity value. An orthogonal matrix is a square matrix where transpose of Square Matrix is also the inverse of Square Matrix.
Orthogonal Matrix in Linear Algebra is a type of matrices in which the transpose of matrix is equal to the inverse of that matrix. As we know, the transpose of a matrix is obtained by swapping its row elements with its column elements. For an orthogonal matrix, the product of the transpose and the matrix itself is the identity matrix, as the transpose also serves as the inverse of the matrix.
Let’s know more about Orthogonal Matrix in detail below.