Solved Examples on Orthogonal Matrix
Example 1: Is every orthogonal matrix symmetric?
Solution:
Every time, the orthogonal matrix is symmetric. Thus, the orthogonal matrix is a property of all identity matrices. An orthogonal matrices will also result from the product of two orthogonal matrices. The orthogonal matrix will likewise have a transpose that is orthogonal.
Example 2: Check whether the matrix X is an orthogonal matrix or not?
[Tex]\bold{\begin{bmatrix} \cos x & \sin x\\ -\sin x & \cos x \end{bmatrix}} [/Tex]
Solution:
We know that the orthogonal matrix’s determinant is always ±1.
The determinant of X = cos x · cos x – sin x · (-sin x)
⇒ |X| = cos2x + sin2x = 1
⇒ |X| = 1
Hence, X is an Orthogonal Matrix.
Example 3: Prove orthogonal property that multiplies the matrix by transposing results into an identity matrix if A is the given matrix.
Solution:
[Tex]A = \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix} [/Tex]
Thus, [Tex]A^{T} = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix} [/Tex]
⇒ [Tex]A \cdot A^{T} = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} [/Tex]
Which is an identity matrix.
Thus, A is an Orthogonal Matrix.
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.