The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$

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 $\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}$.

Proof.

We have
\begin{align*}
(I+A)\left(\, I-\frac{1}{1+\tr(A)}A \,\right)&=I-\frac{1}{1+\tr(A)}A+A-\frac{1}{1+\tr(A)}A^2\6pt] &=I-\frac{1}{1+\tr(A)}\left(\, A-(1+\tr(A))A +A^2\,\right)\\[6pt] &=I-\frac{1}{1+\tr(A)}\left(\, A^2-\tr(A)A \,\right) \tag{*}. \end{align*} The Cayley-Hamilton theorem for 2\times 2 matrices yields that \[A^2-\tr(A)A+\det(A)I=O. Since $A$ is singular, we have $\det(A)=0$.
Hence it follows that we have
$A^2-\tr(A)A=O,$ and we obtain from (*) that
$(I+A)\left(\, I-\frac{1}{1+\tr(A)}A \,\right)=I.$ Similarly,
$\left(\, I-\frac{1}{1+\tr(A)}A \,\right)(I+A)=I.$

Therefore, we conclude that the inverse matrix of $I+A$ is given by the formula
$(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.$

Find the inverse matrix of $\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}$ using the formula

Now let us find the inverse matrix of $\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}$ using the formula.

We first write
$\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}=I+A,$ where
$A=\begin{bmatrix} 1 & 1\\ 1& 1 \end{bmatrix}.$ Then $A$ is a singular matrix with $\tr(A)=2$.

The formula yields that
\begin{align*}
\begin{bmatrix}
2 & 1\\
1& 2
\end{bmatrix}^{-1}&=(I+A)^{-1}\6pt] &=I-\frac{1}{3}A\\[6pt] &=\frac{1}{3}\begin{bmatrix} 2 & -1\\ -1& 2 \end{bmatrix}. \end{align*} Related Question. There is a similar formula for inverse matrices of certain n\times n matrices, called Sherman-Woodberry formula. See the post ↴ Sherman-Woodbery Formula for the Inverse Matrix for the statement of the Sherman-Woodberry formula and its proof. More from my site • If 2 by 2 Matrices Satisfy A=AB-BA, then A^2 is Zero Matrix Let A, B be complex 2\times 2 matrices satisfying the relation \[A=AB-BA. Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.   Hint. Find the trace of $A$. Use the Cayley-Hamilton theorem Proof. We first calculate the […]
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