Note: If $NM,$ it will be much cheaper to evaluate the right$-$hand side of Equation $1,$ which involves

an inverse of a matrix $MM,$ than the left$-$hand side, which involves an inverse of a matrix $NN.$ A

special case that is commonly used in machine learning is:

$(I+AB{)}^{-1}A=A(I+BA{)}^{-1}$

$10$ Points. Verifying the following Woodbury identity:

$(A+B{D}^{-1}C{)}^{-1}=A^{-1}-A^{-1}B(D+C{A}^{-1}B{)}^{-1}C{A}^{-1}$

Note: Equation $3$ is useful when $A$ is large and diagonal $($hence easy to invert$),$ and $B($or $C)$ is a thin

and tall $($or fat and short$)$ matrix.