Given A = $\begin{bmatrix$}[r] 3 & 0 \ 0 & 4 \end{bmatrix}, B = \begin{bmatrix}[r] a & b \ 0 & c \end{bmatrix} and that AB = A + B; find the values of a, b and c.

Given,

$\Rightarrow AB = A + B \\[1em] \Rightarrow \begin{bmatrix$}[r] 3 & 0 \ 0 & 4 \end{bmatrix}\begin{bmatrix}[r] a & b \ 0 & c \end{bmatrix} = \begin{bmatrix}[r] 3 & 0 \ 0 & 4 \end{bmatrix} + \begin{bmatrix}[r] a & b \ 0 & c \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 3 \times a + 0 \times 0 & 3 \times b + 0 \times c \ 0 \times a + 4 \times 0 & 0 \times b + 4 \times c \end{bmatrix} = \begin{bmatrix}[r] 3 + a & b \ 0 & 4 + c \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 3a & 3b \ 0 & 4c \end{bmatrix} = \begin{bmatrix}[r] 3 + a & b \ 0 & 4 + c \end{bmatrix}

By definition of equality of matrices we get,

3a = 3 + a
⇒ 3a - a = 3
⇒ 2a = 3
⇒ a = $\dfrac{3}{2}$.

3b = b
⇒ b = 0.

4c = 4 + c
⇒ 4c - c = 4
⇒ 3c = 4
⇒ c = $\dfrac{4}{3}$

Hence, a = $\dfrac{3}{2}, b = 0 \text{ and c} = \dfrac{4}{3}$.