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 ${{\text{a}}_{\text{1}}}\text{,}{{\text{a}}_{\text{2}}}\text{,}{{\text{a}}_{3}}\text{,}{{\text{b}}_{1}}\text{,}{{\text{b}}_{\text{2}}}\text{,}{{\text{b}}_{3}}$ .

& \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\\ \end{matrix} \\\ & \begin{matrix} {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\\ \end{matrix} \\\ \end{aligned} \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{aligned} & \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{aligned} \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.