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Chapter 7: Problem 38

Prove that if \(A\) and \(B\) are \(n \times n\) orthogonal matrices, then \(A B\) and \(B A\) are orthogonal.

Short Answer

Expert verified
This exercise establishes that if A and B are orthogonal matrices, then their products AB and BA are also orthogonal matrices. The process to prove this involves showing that the transpose of AB times AB and transpose of BA times BA give the identity matrix, which is the defining property of an orthogonal matrix.

Step by step solution

01

Defining an Orthogonal Matrix

An orthogonal matrix is a square matrix whose rows and columns are orthogonal unit vectors (i.e., orthonormal vectors). This means when we take the transpose of the matrix and multiply it by the matrix, we get an identity matrix. This can be represented as \( AA^T = A^T A = I \). We need to prove this property for the matrices AB and BA given that A and B are orthogonal.
02

Proving AB is Orthogonal

We need to show that the transpose of AB times AB equals the identity matrix. Let's find \( (AB)^T AB \). We know that the transposition reverses the order of multiplication, so \( (AB)^T = B^T A^T \). Multiplying these gives \( B^T A^T AB \). As A and B are orthogonal, we know that \( A^T A = B^T B = I \). So, the expression simplifies to \( B^T IB = B^T B = I \). Hence, AB is orthogonal.
03

Proving BA is Orthogonal

We need to follow a similar process to show BA is orthogonal as well. For \( (BA)^T BA = A^T B^T BA \). Utilizing the fact that \( A^T A = B^T B = I \) we get \( A^T IA = A^T A = I \). Hence, BA is orthogonal as well.

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