# Inverse Covariance Matrix Transformation from CAPM

Beginning with the CAPM model we have (with a risk free rate of 0%):

$$r_i=\beta_i (r_m)+\varepsilon_i$$

with $$\varepsilon_i$$ the diversifiable risks per assets

The variance matrix:

$$\Omega = \beta'\beta \sigma_m^2 + Diag(\sigma_e^2)$$

With $$\sigma_m$$ a constant, $$Diag(\sigma_e^2)$$ an N $$\times$$ N matrix, $$\beta$$ an 1 $$\times$$ N matrix.

Inverting the matrix we get the following result:

$$\Omega^{-1} = Diag(\frac{1}{\sigma_e^2})-\frac{(\frac{\beta}{\sigma_e^2})(\frac{\beta}{\sigma_e^2})'}{\frac{1}{\sigma_m^2}+(\frac{\beta}{\sigma_e^2})'\beta}$$

I don't understand how by using the inverse matrix transformation we find this result.

$$\left(\boldsymbol{A}+\boldsymbol{bc}^T\right)^{-1}=\boldsymbol{A}^{-1}-\frac{\boldsymbol{A}^{-1}\boldsymbol{bc}^T\boldsymbol{A}^{-1}}{1+\boldsymbol{c^TA}^{-1}b}$$