Given `A=[{:(,3,0),(,0,4):}], B=[{:(,a,b),(,0,c):}]` and AB=A+B, find the values of a,b and c.

Answer

To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) given the matrices \( A \) and \( B \) and the equation \( AB = A + B \).

Given:
- \( A = \begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix} \)
- \( B = \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \)

We need to compute \( AB \) and set it equal to \( A + B \).

### Step 1: Calculate \( AB \)

To find \( AB \), we multiply matrix \( A \) by matrix \( B \):

\[ AB = \begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix} \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \]

Calculating each element of the resulting matrix:

- First row, first column:
\[ 3 \cdot a + 0 \cdot 0 = 3a \]

- First row, second column:
\[ 3 \cdot b + 0 \cdot c = 3b \]

- Second row, first column:
\[ 0 \cdot a + 4 \cdot 0 = 0 \]

- Second row, second column:
\[ 0 \cdot b + 4 \cdot c = 4c \]

Thus, we have:
\[ AB = \begin{pmatrix} 3a & 3b \\ 0 & 4c \end{pmatrix} \]

### Step 2: Calculate \( A + B \)

Now, we calculate \( A + B \):

\[ A + B = \begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix} + \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \]

Adding the corresponding elements gives us:
\[ A + B = \begin{pmatrix} 3 + a & 0 + b \\ 0 + 0 & 4 + c \end{pmatrix} = \begin{pmatrix} 3 + a & b \\ 0 & 4 + c \end{pmatrix} \]

### Step 3: Set \( AB = A + B \)

Now we set the two results equal to each other:
\[ \begin{pmatrix} 3a & 3b \\ 0 & 4c \end{pmatrix} = \begin{pmatrix} 3 + a & b \\ 0 & 4 + c \end{pmatrix} \]

### Step 4: Equate corresponding elements

From the equality of matrices, we can derive the following equations:

1. \( 3a = 3 + a \)
2. \( 3b = b \)
3. \( 0 = 0 \) (which is always true)
4. \( 4c = 4 + c \)

### Step 5: Solve for \( a \)

From the first equation:
\[ 3a - a = 3 \implies 2a = 3 \implies a = \frac{3}{2} \]

### Step 6: Solve for \( b \)

From the second equation:
\[ 3b - b = 0 \implies 2b = 0 \implies b = 0 \]

### Step 7: Solve for \( c \)

From the fourth equation:
\[ 4c - c = 4 \implies 3c = 4 \implies c = \frac{4}{3} \]

### Final Values

Thus, the values are:
- \( a = \frac{3}{2} \)
- \( b = 0 \)
- \( c = \frac{4}{3} \)