Proving Orthogonal Matrix with Identity Matrix and Non-Zero Column Vector a

[sherlockjones]

Assume that $$I$$ is the $$3\times 3$$ identity matrix and $$a$$ is a non-zero column vector with 3 components. Show that:
$$I - \frac{2}{| a |^{2}}aa^{T}$$ is an orthogonal matrix?My question is how can one take the determinant of $$a$$ if it is not a square matrix? Is there a flaw in this problem?

Thanks

 

[OlderDan]

I assume you are referring to the $$| a |^{2}$$ and I also assume that is the inner product (dot product) for the vector. It's just a normalization factor

 

[HallsofIvy]

Yes. |a| is not a "determinant", it is the length of the vector a.

 

[Tomsk]

Remember that $aa^{T}$ does NOT equal $a^{T}.a$, the scalar product. Use matrix multiplication. You don't need to find the determinant of anything either.