Question

If A = `[(cos α, sin α),(- sin α, cos α)]`, then the matrix A is ______.

Options

• symmetric matrix

• skew-symmetric matrix

• identity matrix

• orthogonal matrix

Answer 1

If A = `[(cos α, sin α),(- sin α, cos α)]`, then the matrix A is orthogonal matrix.

Explanation:

Here, A = `[(cos α, sin α),(- sin α, cos α)]`

Now, transpose of A

A' = `[(cos α, - sin α),(sin α, cos α)]`

A'.A = `[(cos α, - sin α),(sin α, cos α)][(cos α, sin α),(- sin α, cos α)]`

= `[(cos^2 α + sin^2 α, cos α . sin α - sin α . cos α),(sin α . cos α - cos α . sin α, sin^2 α + cos^2 α)]`

= `[(1, 0),(0, 1)]`

= I

So, A is an orthogonal matrix.