Given that A = $\begin{bmatrix} 3 & 0 \ 0 & 2 \end{bmatrix}$, B = $\begin{bmatrix} a & b \ 0 & c \end{bmatrix}$ and AB = (A + B), find the values of a, b, and c.

Solving AB:

$\Rightarrow \begin{bmatrix} 3 & 0 \ 0 & 2 \end{bmatrix} \times \begin{bmatrix} a & b \ 0 & c \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix} (3)(a) + (0)(0) & (3)(b) + (0)(c) \ (0)(a) + (2)(0) & (0)(b) + (2)(c) \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix} 3a & 3b \ 0 & 2c \end{bmatrix}.$

Solving A + B:

$\Rightarrow \begin{bmatrix} 3 & 0 \ 0 & 2 \end{bmatrix} + \begin{bmatrix} a & b \ 0 & c \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix} 3 + a & b \ 0 & 2 + c \end{bmatrix}.$

Given,

AB = A + B

$\Rightarrow \begin{bmatrix} 3a & 3b \ 0 & 2c \end{bmatrix} = \begin{bmatrix} 3 + a & b \ 0 & 2 + c \end{bmatrix}$

∴ 3a = 3 + a

⇒ 3a - a = 3

⇒ 2a = 3

⇒ a = $\dfrac{3}{2}$

∴ 3b = b

⇒ 3b - b = 0

⇒ 2b = 0

⇒ b = 0

∴ 2c = 2 + c

⇒ 2c - c = 2

⇒ c = 2.

Hence, values of a = $\dfrac{3}{2}$, b = 0 c = 2.