Question

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

Answer 1

Hint: 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 $m \times 1$ .
 \[\therefore A = \left[ {\begin{array}{*{20}{c}}
  a \\
  b \\
  c
\end{array}} \right]\]
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, $A = \left[ {\begin{array}{*{20}{c}}
  5&7
\end{array}} \right]$ .