Answer
See below
Work Step by Step
a) Since $A$ is a $1\times 3$ dimension, then $A^T$ will be $3 \times 1$ dimension. $B$ is a $3\times 2$ dimension, then $B^T$ will be $2 \times 3$ dimension. Hence, $B^TA^T$ is a defined expression.
$B^TA^T=\begin{bmatrix}
0 &-7 &-1\\-4 & 1 & -3
\end{bmatrix}\begin{bmatrix}
-3 \\-1 \\6
\end{bmatrix}=\begin{bmatrix}
1 \\ -7
\end{bmatrix}$
b) $C$ is a $2\times 4$ dimension, then $B^T$ will be $4 \times 2$ dimension. Hence, $C^TB^T$ is defined expression.
$C^TB^T=\begin{bmatrix}
-9 &1 \\0 & 1 \\3 &5\\-2 & -2
\end{bmatrix}\begin{bmatrix}
0 &-7 &-1\\-4 & 1 & -3
\end{bmatrix}=\begin{bmatrix}
-4&64&6 \\ -4 &1&-3\\-20 & -16 & -18\\8 & 12 & 8
\end{bmatrix}$
c) We have $D$ is a $3\times 3$ dimension, $D^T$ will be $3 \times 3$ dimension. Since $A$ is $1\times 3$ dimension, hence $D^TA$ is not defined expression.
