orthogonal matrix

Question

Is the determinent of both the orthogonal and orthonormal is +-1?for orthonormal im always getting +-1 but for orthogonal its always +-C,not 1,C=mag of the vector matrix

Comments

Can you give an example to justify your answer?

Answer 1

For an orthogonal matrix, we know

AA^T=I

Now we take determinants on both sides.

|AA^T|=|I|

Determinant of Identity Matrix is 1 and by the property of determinant we can write the above expression as

|A|.|A^T|=1

Another property of orthogonal matrix is, it must be symmetric. Hence A^T = A. Subsisting in the above equation, we get

|A|*|A|=1

|A|^2=1

Therefore det (A) = 1 or -1.