Question

Question: Let U and V be \(n \times n\) orthogonal matrices. Explain why UV is an orthogonal matrix. (That is, explain why UVis invertible and its inverse is \({\left( {UV} \right)^T}\).)

Short answer

It is proved that \(UV\) represents an orthogonal matrix.

Step-by-step solution

Write the orthogonal matrix

If matrix A is an orthogonal matrix, and invertible then it should be a square matrix such that; \({A^{ - 1}} = {A^T}\).

Write the relation for matrices U and V using orthogonal properties

As both U and V are orthogonal matrices, so \({U^{ - 1}} = {U^T}\) and \({V^{ - 1}} = {V^T}\).

Also, \({U^T}U = {U^{ - 1}}U = I\) and \({V^T}V = {V^{ - 1}}V = I\).

Prove that UV is an orthogonal matrix

From the above steps, U and V are invertible, so the product UV must be invertible as:

\begin{gathered} {\left( {UV} \right)^{ - 1}} = {V^{ - 1}}{U^{ - 1}} \\ = {V^T}{U^T} \\ = {\left( {UV} \right)^T}{\text{ }}\left\{ {\because {\text{Theorem 3}}} \right\} \\ \end{gathered}

Thus, \(UV\) represents an orthogonal matrix.