Show that if A is an orthogonal matrix, then A^T is also orthogonal.

Name: Problem 249, Chapter 10: Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry
Uploaded: 2021-09-16T16:55:59-08:00
Duration: 2 min 6 s
Channel: Urvashi Arora
Description: Show that if A is an orthogonal matrix, then A^T is also orthogonal.

step 1 Hello, so the question is taken from determinants and matrices and the question is show that if

Show that if A is an orthogonal matrix, then A^T is also...

Key Concepts

Matrix Inverse

For any invertible matrix, an inverse exists such that A?¹A = I. In the case of orthogonal matrices, the definition A?A = I implies that the inverse of the matrix is its transpose, i.e., A?¹ = A?. This relationship is central to demonstrating that the transpose of an orthogonal matrix is also orthogonal.

Identity Matrix

The identity matrix, denoted as I, is a square matrix with ones on the diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix multiplication. In the context of orthogonal matrices, verifying that A?A = I confirms that A preserves the norm of vectors and that A? acts as the inverse of A.

Orthogonal Matrix

An orthogonal matrix is a square matrix whose columns (and rows) form an orthonormal set. This means that the matrix preserves lengths and angles, and it satisfies the relation A?A = I, where I is the identity matrix. This property is fundamental in understanding rotations and reflections in linear algebra.

Matrix Transpose

The transpose of a matrix is obtained by flipping the matrix over its diagonal, meaning that the row and column indices are swapped. Key properties of the transpose, such as (AB)? = B?A? and (A?)? = A, are essential when proving statements about orthogonal matrices, especially when verifying that the transpose of an orthogonal matrix also maintains the relation necessary to be orthogonal.

Transcript

Please give Ace some feedback