Question

Suppose that θ is some angle. Consider the following matrix A =[ cos(θ) − sin(θ) sin(θ) cos(θ)]

a) Prove that, for every angle θ, A is an orthogonal matrix.

Answer 1

To prove matrix A is orthogonal, we showed the product of matrix A and its transpose equals the identity matrix by using properties of trigonometric functions, thus confirming that matrix A is orthogonal for every angle θ.

Step-by-step explanation:

To prove that matrix A = [ cos(θ) -sin(θ) sin(θ) cos(θ)] is an orthogonal matrix for every angle θ, we need to show that A multiplied by its transpose AT equals the identity matrix I. The transpose AT of matrix A is [ cos(θ) sin(θ) -sin(θ) cos(θ)].

Now, let's compute the product of A and AT:

• Top left: cos(θ) * cos(θ) + (-sin(θ)) * sin(θ) = cos²(θ) + sin²(θ) = 1,
• Top right: cos(θ) * sin(θ) + (-sin(θ)) * cos(θ) = 0,
• Bottom left: sin(θ) * cos(θ) + cos(θ) * (-sin(θ)) = 0,
• Bottom right: sin(θ) * sin(θ) + cos(θ) * cos(θ) = sin²(θ) + cos²(θ) = 1.

Since the product of A and its transpose AT results in the identity matrix with 1's on the diagonal and 0's elsewhere, we can conclude that matrix A is indeed orthogonal for any θ.