Alright, let's solve this step by step.
Given the matrices:
[tex]\[ A = \begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \][/tex]
We are given that:
[tex]\[ AB = A + B \][/tex]
First, let's compute [tex]\( AB \)[/tex]:
[tex]\[ AB = \begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix} \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \][/tex]
Multiplying these matrices:
[tex]\[ AB = \begin{pmatrix} 3a + 0 \cdot 0 & 3b + 0 \cdot c \\ 0 \cdot a + 4 \cdot 0 & 0 \cdot b + 4c \end{pmatrix} = \begin{pmatrix} 3a & 3b \\ 0 & 4c \end{pmatrix} \][/tex]
Now let's compute [tex]\( A + B \)[/tex]:
[tex]\[ A + B = \begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix} + \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} = \begin{pmatrix} 3 + a & b \\ 0 & 4 + c \end{pmatrix} \][/tex]
According to the problem, [tex]\( AB \)[/tex] must equal [tex]\( A + B \)[/tex]. So we set these two matrices equal:
[tex]\[ \begin{pmatrix} 3a & 3b \\ 0 & 4c \end{pmatrix} = \begin{pmatrix} 3 + a & b \\ 0 & 4 + c \end{pmatrix} \][/tex]
Equating the corresponding elements, we get the following system of equations:
1. [tex]\( 3a = 3 + a \)[/tex]
2. [tex]\( 3b = b \)[/tex]
3. [tex]\( 4c = 4 + c \)[/tex]
Let's solve these equations one by one:
1. For [tex]\( 3a = 3 + a \)[/tex]:
[tex]\[ 3a - a = 3 \][/tex]
[tex]\[ 2a = 3 \][/tex]
[tex]\[ a = \frac{3}{2} \][/tex]
2. For [tex]\( 3b = b \)[/tex]:
[tex]\[ 3b - b = 0 \][/tex]
[tex]\[ 2b = 0 \][/tex]
[tex]\[ b = 0 \][/tex]
3. For [tex]\( 4c = 4 + c \)[/tex]:
[tex]\[ 4c - c = 4 \][/tex]
[tex]\[ 3c = 4 \][/tex]
[tex]\[ c = \frac{4}{3} \][/tex]
Therefore, the values are:
[tex]\[ a = \frac{3}{2}, \][/tex]
[tex]\[ b = 0, \][/tex]
[tex]\[ c = \frac{4}{3}. \][/tex]
These values satisfy the condition [tex]\( AB = A + B \)[/tex].
