Question

The transpose of a square matrix is a:
(a) Rectangular matrix
(b) Diagonal matrix
(c) Square matrix
(d) Scalar matrix

Answer 1

Hint: In the transpose of a given matrix rows become columns and columns become rows. We are asked to find the transpose of a square matrix as in the square matrix number of rows and columns are equal so after taking the transpose of a square matrix then the number of rows and columns are the same so the matrix that we got is the square matrix.

Complete step-by-step answer:
We are asked to find the transpose of a square matrix. Let us assume a $3\times 3$ square matrix which is given below and let us name it as “A” matrix.
$A=\left( \begin{matrix}
   {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\
   {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\
   {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\
\end{matrix} \right)$
We know that when we take the transpose of any matrix then rows become columns and columns become rows so taking the transpose of the above matrix the first row becomes the first column of the new matrix. Similarly the second and third row becomes second and third column respectively. We are denoting the transpose of the above matrix as ${{A}^{T}}$.
${{A}^{T}}=\left( \begin{matrix}
   {{a}_{11}} & {{a}_{21}} & {{a}_{31}} \\
   {{a}_{12}} & {{a}_{22}} & {{a}_{32}} \\
   {{a}_{13}} & {{a}_{23}} & {{a}_{33}} \\
\end{matrix} \right)$
From the transpose of the above matrix we can see that the matrix that we have got is the square matrix of order $3\times 3$.
From the above, we have found that the transpose of a square matrix is a square matrix.
Hence, the correct option is (c).

Note: You might be inquisitive why other options could not be possible. Now, in the below we are going to discuss why the other options could not be possible.
Option (a) is the rectangular matrix. This option is straight away wrong because we have got the transpose as a square matrix so rectangular could not be possible.
Option (b) is the diagonal matrix. You might think that this option could be the correct answer because the diagonal matrix is also a square matrix but in the diagonal matrix all the elements apart from the diagonal are 0. If we take the transpose of the matrix then elements other than diagonal not becomes 0 due to this fact this option is incorrect.
Option (c) is the correct answer which we have already explained above.
Option (d) is the scalar matrix. As a scalar matrix is the form of a diagonal matrix, the only difference is that the elements in the diagonal of the scalar matrix are not 1 which is with the case in diagonal matrix. But in option (b) we have shown that the diagonal matrix could not be the correct choice then this automatically makes the option (d) as incorrect.