Question

Matrix $\left[ \begin{matrix} 0 \\4 \\ -1 \end{matrix} \begin{matrix} -4 \\ 0 \\ 5 \end{matrix} \begin{matrix} 1 \\ 5 \\ 0 \end{matrix} \right] $ is
A. Orthogonal
B. Idempotent
C. Skew-Symmetric
D. Symmetric

Answer 1

Hint:
Firstly we are given a matrix A and we have to find its transpose and then after finding the transpose we will see that it will be equal to the negative of the given matrix and we will get our required solution.


Formula Used:
Let we are given a matrix
A=\[\left [ {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\{{a_{31}}}&{{a_{32}}}&{{a_{33}}}\end{array}} \right]\]. Now we will find its transpose, so for that its transpose is \[\left[{\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\{{a_{31}}}&{{a_{32}}}&{{a_{33}}}\end{array}} \right]\]and then after finding it we will see that it will be equal to its negative of the given matrix.

Complete Step-by-Step Solution:
Now we will see whether this matrix is Skew-Symmetric or not
A skew-symmetric matrix is a square matrix that is equal to the negative of its transpose matrix. It is important to know the method to find the transpose of a matrix, in order to understand a skew-symmetric matrix better. Here, we have considered a matrix A. The basic formula representing a Skew Symmetric Matrix is as follows.
\[A = - {A^T}\]
Now for matrix \[ \left[ \begin{matrix} 0 \\4 \\ -1 \end{matrix} \begin{matrix} -4 \\ 0 \\ 5 \end{matrix} \begin{matrix} 1 \\ 5 \\ 0 \end{matrix} \right] \] we will firstly find its transpose
So \[{A^T} = \left[ {\begin{array}{*{20}{c}}0&{ 4}&-1\\{-4}&0&{ 5}\\{ 1}&{-5}&0\end{array}} \right]\] as we interchange rows to columns and vice versa
Now we can see that this \[{A^T} = - A\].
Hence it follows the definition of a skew-symmetric matrix.
Option C is correct.


Note:
Students should always remember the definitions of such matrices as in such questions it will be very helpful to them. And whenever such a question comes, we can transpose the matrix and then in the next step we will see which type of matrix it will become.