# Tagged: Sherman-Woodberry formula

## Problem 505

Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.
Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:
$(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.$

Using the formula, calculate the inverse matrix of $\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}$.

## Problem 250

Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies
$\mathbf{v}^{\trans}\mathbf{u}\neq -1.$ Define the matrix
$A=I+\mathbf{u}\mathbf{v}^{\trans}.$

Prove that $A$ is invertible and the inverse matrix is given by the formula
$A^{-1}=I-a\mathbf{u}\mathbf{v}^{\trans},$ where
$a=\frac{1}{1+\mathbf{v}^{\trans}\mathbf{u}}.$ This formula is called the Sherman-Woodberry formula.