Question

If matrices Aand Bcommute, then matrices ${\mathit{A}}^{\mathbf{T}}$and ${\mathit{B}}^{\mathbf{T}}$must commute as well.

Short answer

True

Step-by-step solution

Step by step solution Step 1: Properties of the transpose

Consider the properties of the transpose below.

1. $\mathbf{(}\mathbf{A}\mathbf{+}\mathbf{B}{\mathbf{)}}^{\mathbf{T}}\mathbf{=}{\mathit{A}}^{\mathbf{T}}\mathbf{+}{\mathit{B}}^{\mathbf{T}}$ For all m×nmatrices Aand B.
2. $\mathbf{(}\mathbf{k}\mathbf{A}{\mathbf{)}}^{\mathbf{T}}\mathbf{=}\mathit{k}{\mathit{A}}^{\mathbf{T}}$ For all m×nmatrices Aand for all scalars k.
3. $\mathbf{(}\mathbf{A}\mathbf{B}{\mathbf{)}}^{\mathbf{T}}\mathbf{=}{\mathit{B}}^{\mathbf{T}}{\mathit{A}}^{\mathbf{T}}$ For all m×pmatrices Aand for all matrices B.
4. $\mathbf{\text{rank}}\mathbf{\left(}{\mathit{A}}^{\mathbf{T}}\mathbf{\right)}\mathbf{=}\mathbf{\text{rank}}\mathbf{\left(}\mathit{A}\mathbf{\right)}$ For all matrices A.
5. $\mathbf{(}{\mathbf{A}}^{\mathbf{T}}{\mathbf{)}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\mathbf{(}{\mathbf{A}}^{\mathbf{-}\mathbf{1}}{\mathbf{)}}^{\mathbf{T}}$ For all invertible n×nmatrices A.

Determine whether the statement is true or false

By the above theorem, consider the terms below.

$\begin{array}{rcl}{A}^T{B}^T& =& (BA{)}^T\\ & =& (AB{)}^T\\ & =& B^T{A}^T\end{array}$

Hence, the statement is true.