I'm having trouble simplifying the following expression in matrix form:
$$\mathbf{X}(\mathbf{X}+a\mathbf{I})^{-1}$$
Where $\mathbf{X}$ is an invertible $n \times n$ matrix, $a$ is a scalar value, and $\mathbf{I}$ is the identity matrix.
I reasoned that since the product of a matrix times its inverse is the identity matrix, the product of a matrix times the inverse of a "shifted" matrix is simply the shift value. In other words, the above would simplify to $a\mathbf{I}$. However, I don't know if that is correct, and if it is I don't know the linear algebra steps to prove it.
$X(X+aI)^{-1} = ((X+aI) X^{-1})^{-1} = (I+ a X^{-1})^{-1}$.