Question

The transpose of a column matrix is
A.Zero matrix
B.Diagonal matrix
C.Column matrix
D.Row matrix

Answer 1

Consider any matrix A which is a column matrix of order $m \times 1$ .
Then, do the transpose of the matrix A and find ${A^T}$ .
Now, check what type of matrix is ${A^T}$ and decide its type.
Transpose of a matrix:
Transpose of a matrix is an operator which switches the rows and columns of a matrix A by forming a new matrix which is denoted by ${A^T}$ .

Complete step-by-step answer:
Let A be a column matrix of order $m \times 1$ .

a \\\ b \\\ c \end{array}} \right]$$ Now, we are asked to do the transpose of the matrix. When we transpose any matrix of order $m \times n$ , its transpose will have the order $n \times m$ . So, here on transposing A, ${A^T} = \left[ {\begin{array}{*{20}{c}} a&b;&c; \end{array}} \right]$ , which is of order $1 \times m$ . Also, any matrix of order $1 \times m$ is a row matrix. Thus, from ${A^T} = \left[ {\begin{array}{*{20}{c}} a&b;&c; \end{array}} \right]$ and its order $1 \times m$ , we get the transpose of a column matrix as a row matrix. **So, option (D) Row matrix is correct.** **Note:** Zero matrix: Any square matrix A of order $m \times m$ , where all the elements of the matrix have value 0, is called a zero matrix. For example, $A = \left[ {\begin{array}{*{20}{c}} 0&0 \\\ 0&0 \end{array}} \right]$ Diagonal matrix: Any square matrix A of order $m \times m$ , where elements expect the elements of primary diagonal are 0 is called a diagonal matrix. The elements of primary diagonal are ${a_{ij}},i = j$ . For example, $A = \left[ {\begin{array}{*{20}{c}} 2&0 \\\ 0&5 \end{array}} \right]$ . Column matrix: Any matrix A of the order $m \times 1$ is called a column matrix. For example, $A = \left[ {\begin{array}{*{20}{c}} 1 \\\ 2 \end{array}} \right]$ . Row matrix: Any matrix A of the order $1 \times n$ is called a row matrix. For example, $A = \left[ {\begin{array}{*{20}{c}} 5&7 \end{array}} \right]$ .