Identity matrix addition and inverse matrices

Question

I am trying to reduce the following:

x and y column vectors yt is the transposed column vector

$(I - \frac{1}{(1+ y^t x)} * x y^t) (I + x y^t) = I$

I am stuck at $x y^t * y^t X = x y^t (x y^t +I)$

How do I reduce it further?

Answer 1

$xy^Txy^T=x(y^Tx)y^T=(y^Tx)*xy^T$ by associativity and then converting the 1-by-1 matrix to a scalar. So you get many scalar multiples of $xy^T$ which cancel out:

$(I-{1\over 1+y^Tx}xy^T)(I+xy^T)$
$=I+xy^T-{1\over 1+y^Tx}*(1*xy^T+(y^Tx)*xy^T)$
$=I+xy^T-xy^T=I$.

So the equation is an identity.