If Q is an orthogonal matrix, what is Q-1?

Use the defining property. By definition, an orthogonal matrix satisfies QTQ = I. Identify the inverse. This identity says QT is a two-sided inverse of Q.

Inverse of an Orthogonal Matrix

Linear Algebra

Orthogonal Matrices

Easy

Q

is an orthogonal matrix, what is

Q^{- 1}

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Hint 1

An orthogonal matrix satisfies $Q^{T} Q = I$ by definition.

Hint 2

This identity directly identifies the inverse.

Show worked solution

1. Use the defining property
By definition, an orthogonal matrix satisfies $Q^{T} Q = I$ .
2. Identify the inverse
This identity says $Q^{T}$ is a two-sided inverse of $Q$ .
$Q^{- 1} = Q^{T}$

Answer: Q^T

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Inverse of an Orthogonal Matrix

1. Use the defining property

2. Identify the inverse

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