Question

The transpose of a rectangular matrix is a
A) Rectangular matrix
B) Diagonal matrix
C) Square matrix
D) Scalar matrix

Answer 1

Hint: The transpose of a matrix is simply a flipped version of the original matrix. We can get the transpose of a matrix by exchanging its rows with its columns. Suppose a matrix A, then its transpose is denoted as \[{{\text{A}}^{\text{T}}}\] , where the superscript T means transpose of the matrix. Assume a matrix having \[2\times 3\] dimensions and then find its transpose.
Complete step-by-step answer:
Let us assume a \[2\times 3\] rectangular matrix A whose elements are \[{{\text{a}}_{\text{1}}}\text{,}{{\text{a}}_{\text{2}}}\text{,}{{\text{a}}_{3}}\text{,}{{\text{b}}_{1}}\text{,}{{\text{b}}_{\text{2}}}\text{,}{{\text{b}}_{3}}\] .
\[A=\left[ \begin{align}
  & \begin{matrix}
   {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\
\end{matrix} \\
 & \begin{matrix}
   {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\
\end{matrix} \\
\end{align} \right]\] …………………….(1)
We know that the transpose of a matrix is given by exchanging its rows with its columns.
Exchanging the rows and columns of the matrix of equation (1), we get
\[{{A}^{T}}=\left[ \begin{align}
  & \begin{matrix}
   {{a}_{1}} & {{b}_{1}} \\
\end{matrix} \\
 & \begin{matrix}
   {{a}_{2}} & {{b}_{2}} \\
\end{matrix} \\
 & \begin{matrix}
   {{a}_{3}} & {{b}_{3}} \\
\end{matrix} \\
\end{align} \right]\] ……………………(2)
We know that a rectangular matrix has either \[m\times n\] dimensions or \[n\times m\] dimensions.
So, both \[2\times 3\] matrix and \[3\times 2\] matrix are rectangular.
We can see that the matrix of equation (2) is rectangular.
Hence, the transpose of a rectangular matrix is also a rectangular matrix.
Therefore, option (A) is the correct one.

Note: We can also solve this question by using the property that if a matrix has \[m\times n\] dimensions then the transpose of that matrix will have dimensions \[n\times m\] .
Assume a rectangular matrix having \[m\times n\] dimensions. Then, its transpose will have \[n\times m\] dimensions.
We know that a rectangular matrix has either \[m\times n\] dimensions or \[n\times m\] dimensions.
Therefore, the matrix having \[n\times m\] dimensions is also rectangular.
Hence, the transpose of a rectangular matrix is also rectangular.