Question

IfA andB are orthogonal$\mathit{n}\mathbf{\times }\mathit{n}$ matrices such that$\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathit{B}\mathbf{=}\mathbf{1}$, then matricesA and Bmust commute.

Short answer

Therefore,

$AB\ne BA$

So, the given statement is false.

Step-by-step solution

Orthogonal Matrix Definition

A square matrix with real numbers or elements is said to be an orthogonal matrix, if its transpose is equal to its inverse matrix.

Or we can say, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix

To check whether the given condition is true or false

For example,

$A=\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]$

$B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right]$

Clearly, bothA andB are orthogonal matrices, and$detA=detB=1$.

However,

$AB=\left[\begin{array}{ccc}0& 0& 1\\ -1& 0& 0\\ 0& -1& 0\end{array}\right]$

$AB\ne \left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$

$AB\ne BA$.

So, the given statement is false.