The inverse of an orthogonal matrix is its transpose

The inverse of the orthogonal matrix is also orthogonal. It is the matrix product of two matrices that are orthogonal to each other. If inverse of the matrix is equal to its transpose, then it is an orthogonal matrix.

1. Why is the transpose of an orthogonal matrix orthogonal?
2. What is an orthogonal matrix times its transpose?
3. Why is an orthogonal matrix invertible?
4. Is invertible the same as transpose?

Why is the transpose of an orthogonal matrix orthogonal?

As mentioned above, the transpose of an orthogonal matrix is also orthogonal. In fact its transpose is equal to its multiplicative inverse and therefore all orthogonal matrices are invertible.

What is an orthogonal matrix times its transpose?

An orthogonal matrix multiplied with its transpose is equal to the identity matrix.

Why is an orthogonal matrix invertible?

An orthogonal matrix is invertible by definition, because it must satisfy ATA=I. In an orthogonal matrix the columns are pairwise orthogonal and each is a norm 1 vector, so they form an orthonormal basis.

Is invertible the same as transpose?

The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix. The notation A^−T is sometimes used to represent either of these equivalent expressions.