Question

The Matrix \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 0&{ - 4}&1\\ 4&0&{ - 5}\\ { - 1}&5&0 \end{array}} \right]\) is
1. Orthogonal Matrix
2. Skew Symmetric
3. Symmetric
4. Idempotent

Answer 1

Correct Answer - Option 2 : Skew Symmetric

Concept:

Skew-symmetric or antisymmetric matrix is a square matrix whose transpose equals its negative, that is, it satisfies the condition A^T = −A.

Here, the matrix A is, \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 0&{ - 4}&1\\ 4&0&{ - 5}\\ { - 1}&5&0 \end{array}} \right]\)

The transpose of matrix A is, \({\rm{A}^T} = \left[ {\begin{array}{*{20}{c}} 0&4&{ - 1}\\ { - 4}&0&5\\ 1&{ - 5}&0 \end{array}} \right]\)

As A^T ≠ A, the matrix is not symmetric.

\(\therefore {\rm{A}^T} = \left( { - 1} \right)\left[ {\begin{array}{*{20}{c}} 0&{ - 4}&1\\ 4&0&{ - 5}\\ { - 1}&5&0 \end{array}} \right]\)

A^T = (-1) A

A^T = - A

Since A^T = -A, hence matrix is skew-symmetric.