Find the matrix B if A = $\begin{bmatrix$}[r] 4 & 1 \ 2 & 3 \end{bmatrix} and A² = A + 2B.

Given, A² = A + 2B.

$\Rightarrow \begin{bmatrix$}[r] 4 & 1 \ 2 & 3 \end{bmatrix} \begin{bmatrix}[r] 4 & 1 \ 2 & 3 \end{bmatrix} = \begin{bmatrix}[r] 4 & 1 \ 2 & 3 \end{bmatrix} + 2B \\[0.5em] \Rightarrow \begin{bmatrix}[r] 4 \times 4 + 1 \times 2 & 4 \times 1 + 1 \times 3 \ 2 \times 4 + 3 \times 2 & 2 \times 1 + 3 \times 3 \end{bmatrix} = \begin{bmatrix}[r] 4 & 1 \ 2 & 3 \end{bmatrix} + 2B \\[0.5em] \Rightarrow \begin{bmatrix}[r] 16 + 2 & 4 + 3 \ 8 + 6 & 2 + 9 \end{bmatrix} = \begin{bmatrix}[r] 4 & 1 \ 2 & 3 \end{bmatrix} + 2B \\[0.5em] \Rightarrow \begin{bmatrix}[r] 18 & 7 \ 14 & 11 \end{bmatrix} = \begin{bmatrix}[r] 4 & 1 \ 2 & 3 \end{bmatrix} + 2B \\[0.5em] \Rightarrow 2B = \begin{bmatrix}[r] 18 & 7 \ 14 & 11 \end{bmatrix} - \begin{bmatrix}[r] 4 & 1 \ 2 & 3 \end{bmatrix} \\[0.5em] \Rightarrow 2B = \begin{bmatrix}[r] 18 - 4 & 7 - 1 \ 14 - 2 & 11 - 3 \end{bmatrix} \\[0.5em] \Rightarrow 2B = \begin{bmatrix}[r] 14 & 6 \ 12 & 8 \end{bmatrix} \\[0.5em] \therefore B = \begin{bmatrix}[r] 7 & 3 \ 6 & 4 \end{bmatrix}.

Hence, the matrix B = $\begin{bmatrix$}[r] 7 & 3 \ 6 & 4 \end{bmatrix}.