Question on Identity Matrix

Question

Suppose $B$ is a $2 \times 2$ matrix satisfying $AB = B+I$, where $$A = \begin{bmatrix}4&2\\2&3\end{bmatrix}$$ Find B.

Answer 1

$$AB = B+I \implies AB - B = I \implies AB - IB = I \implies (A-I)B=I \\ \implies (A-I)^{-1} = B$$

Answer 2

$AB=B+I$

-> $(A-I)B=I$

$A-I=\begin{bmatrix}3&2\\2&2\end{bmatrix}$

$B=\begin{bmatrix}1&-1\\-1&\frac{3}{2}\end{bmatrix}$

Answer 3

A summary of the other answers. Hopefully this clears things up.

\begin{equation} \begin{split} AB = B+I &\iff AB - B = I \\ &\iff AB - IB = I \\ &\iff (A-I)B=I. \end{split} \end{equation} $\det(A - I) =6-4=2\neq 0$, and therefore $A-I$ has an inverse. Hence $$B = (A-I)^{-1}.$$