Question

TRUE OR FALSE?The equation${\left(AB\right)}^{\mathbf{T}}\mathbf{=}{\mathit{A}}^{\mathbf{T}}{\mathit{B}}^{\mathbf{T}}$ holds for all $\mathbf{n}\mathbf{\times }\mathbf{n}$, matricesA andB.

Short answer

The given statement is false

Step-by-step solution

Recollect the properties of the transpose of a matrix.

According to the properties of a transpose of a $\mathit{n}\mathbf{\times }\mathit{n}$matrix, for any Aand Bmatrices, ${\mathit{\left(}\mathit{A}\mathit{B}\mathit{\right)}}^{\mathit{T}}\mathit{=}{\mathit{B}}^{\mathit{T}}{\mathit{A}}^{\mathit{T}}$.

Check whether the given statement is a true or false.

The given statement is The equation${\mathit{\left(}AB\mathit{\right)}}^T\mathit{=}{A}^T{B}^T$ holds for all$n\times n$ matricesA and B.

It is given that the vectors${\mathit{\left(}AB\mathit{\right)}}^T\mathit{=}{A}^T{B}^T$ .

But, by the above stated property,${\mathit{\left(}AB\mathit{\right)}}^T\mathit{=}{B}^T{A}^T$ for all$n\times n$ matricesA and B.

It is know that the matrix multiplication is not commutative.

So, it is clear that $A^{\mathit{T}}{B}^{\mathit{T}}\mathit{=}{B}^T{A}^T$is cannot be true for every case.

Then the given statement is false, by the properties of the transpose of matrices.