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Given the matrix A_α = [cos α -sin α; sin α cos α], which of the following identities is true?

1. A_α · A_β = I
2. A_α · A_β = O
3. A_α · A_β = A_{α+β}
4. A_α · A_β = A_{α-β}

Correct answer: A_α · A_β = A_{α+β}

Solution

The correct option states that the product of two rotation matrices A_α and A_β results in another rotation matrix A_{α+β}, which reflects the additive property of angles in trigonometric functions. This means that rotating by angle α followed by angle β is equivalent to a single rotation by the sum of those angles.

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