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.