Question

If A is a square matrix, then A - A' is a:

1. Diagonal matrix.
2. Skew-symmetric matrix.
3. Symmetric matrix.
4. None of these.

Answer 1

Correct Answer - Option 2 : Skew-symmetric matrix.

Concept:

Transpose of a Matrix:

The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix.

It is denoted by A' or A^T.

 

A symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric if and only if A = A'.

A skew-symmetric matrix is a square matrix that is equal to its transpose. Formally, A is skew-symmetric if and only if A = - A'.

A diagonal matrix is a matrix in which all the entries except the diagonal are 0.

Calculation:

It is given that A is a square matrix. 

Now, transpose of  A - A'

= (A - A')'

= A' - (A')'

= A' - A

= -(A - A')

Since A - A' is equal to the negative of its transpose, the matrix A - A' is skew-symmetric.