Prove that the inverse of an orthogonal matrix is orthogonal.

Name: Problem 7, Chapter 4: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2021-09-16T16:55:59-08:00
Duration: 2 min 6 s
Channel: Urvashi Arora
Description: Prove that the inverse of an orthogonal matrix is orthogonal.

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

Prove that the inverse of an orthogonal matrix is...

Key Concepts

Orthogonal Matrix

An orthogonal matrix is a square matrix whose columns (and rows) form an orthonormal set with respect to the dot product. This means that the matrix multiplied by its transpose yields the identity matrix (A?A = I), indicating that the transformation preserves lengths and angles. This property is essential in diverse applications, including rotations and reflections in Euclidean space.

Matrix Inverse

The inverse of a matrix A, denoted A?¹, is defined such that when it is multiplied with A, it results in the identity matrix (AA?¹ = I). The concept of an inverse matrix is fundamental in linear algebra because it provides a way to 'undo' the transformation represented by A, thereby allowing for the solution of linear systems and the analysis of linear transformations.

Matrix Transpose

The transpose of a matrix is obtained by flipping the matrix over its diagonal, thus switching its rows with its columns. In the context of orthogonal matrices, the transpose plays a pivotal role since for such matrices the transpose is equal to the inverse (A? = A?¹). This relationship is key to understanding the structure and properties of orthogonal transformations.

Identity Matrix

The identity matrix is a special square matrix that acts as the neutral element in matrix multiplication; every matrix multiplied by the identity matrix remains unchanged. In proofs involving inverses and orthogonal matrices, the identity matrix serves as the reference point, demonstrating that the product of a matrix and its inverse (or transpose, in the orthogonal case) yields this neutral element, confirming the matrix's invertibility and orthogonality.

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