Question

What is the transpose of a row matrix is :
A) Zero matrix
B) Diagonal matrix
C) Column matrix
D) row matrix

Answer 1

Hint: Suppose A matrix (A) is there with an order of 3 × 3. Then taking its transpose such that the elements present in a row come under the place of the elements in a column.

Complete step by step solution: Let us assume a matrix “A” of order 3 × 3 such that there are 3 rows and 3 columns :
${\text{A}} = \left( {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\
  {{b_1}}&{{b_2}}&{{b_3}} \\
  {{c_1}}&{{c_2}}&{{c_3}}
\end{array}} \right)$
Then, we will find its transpose, which is indicated by\[({A^T})\]
\[{{\text{A}}^{\text{T}}} = \left( {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{c_1}} \\
  {{a_2}}&{{b_2}}&{{c_2}} \\
  {{a_3}}&{{b_3}}&{{c_3}}
\end{array}} \right)\]Such all the element
like \[{a_1},{a_2},{a_3}\]were present horizontally in a particular row, now becomes the element of 3 particular columns,
Similarly \[{b_1},{b_2},{b_3}\]and \[{c_1},{c_2},{c_{3.}}\]
Hence, Row matrix after finding its transpose becomes column matrix.

So, option (C) column matrix is correct.

Note: In this type of Question first, we need to assume an unknown matrix, then apply the condition which has been asked. And here is given about the matrix and its transpose. So we just transfer the elements in the rows to the column. To find its transpose denoted by \[({A^T})\] where A is the matrix.
If \[{\text{A}} = \left( {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\
  {{b_1}}&{{b_2}}&{{b_3}} \\
  {{c_1}}&{{c_2}}&{{c_3}}
\end{array}} \right){\text{then}}\,{{\text{A}}^{\text{T}}} = \left( {\begin{array}{*{20}{c}}
  {{a_1}}&{{b_1}}&{{c_1}} \\
  {{a_2}}&{{b_2}}&{{c_2}} \\
  {{a_3}}&{{b_3}}&{{c_3}}
\end{array}} \right)\]
And in this way we can solve.