Question

If \[{{A}^{T}}=\left[ \begin{matrix}
   3 & 4 \\
   -1 & 2 \\
   0 & 1 \\
\end{matrix} \right]\] and \[B=\left[ \begin{matrix}
   -1 & 2 & 1 \\
   1 & 2 & 3 \\
\end{matrix} \right]\]. Find the value of \[{{A}^{T}}-{{B}^{T}}\].

Answer 1

Hint: In this question, We are given with \[{{A}^{T}}=\left[ \begin{matrix}
   3 & 4 \\
   -1 & 2 \\
   0 & 1 \\
\end{matrix} \right]\] and \[B=\left[ \begin{matrix}
   -1 & 2 & 1 \\
   1 & 2 & 3 \\
\end{matrix} \right]\]. Now we know that if the dimension of the matrix \[A\] is \[m\times n\], then the dimension of the matrix \[{{A}^{T}}\] is given by \[n\times m\]. Moreover the matrix \[{{A}^{T}}\] is determined by the matrix \[A\] where the corresponding rows of matrix \[A\] becomes the corresponding columns of matrix \[{{A}^{T}}\]. Using this we will determine the matrix \[{{B}^{T}}\] and then finally we will find the matrix \[{{A}^{T}}-{{B}^{T}}\] by subtracting the elements of \[{{B}^{T}}\] from the corresponding elements of \[{{A}^{T}}\] only of dimension of both the matrices are same.

Complete step-by-step answer:
We are given with \[{{A}^{T}}=\left[ \begin{matrix}
   3 & 4 \\
   -1 & 2 \\
   0 & 1 \\
\end{matrix} \right]\] and \[B=\left[ \begin{matrix}
   -1 & 2 & 1 \\
   1 & 2 & 3 \\
\end{matrix} \right]\].
Now we can see there are 3 rows and 2 columns in the matrix \[{{A}^{T}}\].
Thus the dimension of the matrix \[{{A}^{T}}\] is given by \[3\times 2\].

Also since there are 2 rows and 3 columns in the matrix \[B\], thus the dimension of the matrix \[B\] is given by \[2\times 3\].
 Now using the fact that if the dimension of matrix \[A\] is \[m\times n\], then the dimension of the matrix \[{{A}^{T}}\] is given by \[n\times m\].
We have that the dimension of \[{{B}^{T}}\] is given by \[3\times 2\].
That is there are 3 rows and 2 columns in the matrix \[{{B}^{T}}\].
Also since we know that matrix \[{{A}^{T}}\] is determined by the matrix \[A\] where the corresponding rows of matrix \[A\] becomes the corresponding columns of matrix \[{{A}^{T}}\].
We will then determine the matrix \[{{B}^{T}}\] from the matrix \[B\].
Now since \[B=\left[ \begin{matrix}
   -1 & 2 & 1 \\
   1 & 2 & 3 \\
\end{matrix} \right]\], thus we have
\[{{B}^{T}}=\left[ \begin{matrix}
   -1 & 1 \\
   2 & 2 \\
   1 & 3 \\
\end{matrix} \right]\]
Now since the dimension of both the matrices \[{{A}^{T}}\] and \[{{B}^{T}}\] is the same. i.e., \[3\times 2\].
So in order to find the matrix \[{{A}^{T}}-{{B}^{T}}\], we will have to subtract each elements of \[{{B}^{T}}\] from the corresponding elements of \[{{A}^{T}}\].
Thus we get ,

\[\begin{align}
  & {{A}^{T}}-{{B}^{T}}=\left[ \begin{matrix}
   3 & 4 \\
   -1 & 2 \\
   0 & 1 \\
\end{matrix} \right]-\left[ \begin{matrix}
   -1 & 1 \\
   2 & 2 \\
   1 & 3 \\
\end{matrix} \right] \\
 & =\left[ \begin{matrix}
   3-\left( -1 \right) & 4-1 \\
   -1-2 & 2-2 \\
   0-1 & 1-3 \\
\end{matrix} \right] \\
 & =\left[ \begin{matrix}
   4 & 3 \\
   -3 & 0 \\
   -1 & -2 \\
\end{matrix} \right]
\end{align}\]
Hence we finally have \[{{A}^{T}}-{{B}^{T}}=\left[ \begin{matrix}
   4 & 3 \\
   -3 & 0 \\
   -1 & -2 \\
\end{matrix} \right]\].

Note: In this problem, take care of the fact that dimensions of matrices have to be the same in order to add them or subtract them. In this case the dimension of the matrices \[{{A}^{T}}\] and \[B\] are not the same. Hence we cannot add or subtract them.