If `[{:(,a,3),(,4,1):}]+[{:(,2,b),(,1,-2):}]-[{:(,1,1),(,-2,c):}] =[{:(,5,0),(,7,3):}]`
find the values of a,b and c.

Answer

To solve the equation involving matrices, we start with the given expression:

\[ \begin{pmatrix} a & 3 \\ 4 & 1 \end{pmatrix} + \begin{pmatrix} 2 & b \\ 1 & -2 \end{pmatrix} - \begin{pmatrix} 1 & 1 \\ -2 & c \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 7 & 3 \end{pmatrix} \]

### Step 1: Combine the matrices on the left side
We will first combine the matrices on the left side. This involves adding the first two matrices and then subtracting the third matrix.

1. **Addition of the first two matrices:**
\[ \begin{pmatrix} a + 2 & 3 + b \\ 4 + 1 & 1 - 2 \end{pmatrix} = \begin{pmatrix} a + 2 & 3 + b \\ 5 & -1 \end{pmatrix} \]

2. **Now subtract the third matrix:**
\[ \begin{pmatrix} a + 2 - 1 & 3 + b - 1 \\ 5 - (-2) & -1 - c \end{pmatrix} = \begin{pmatrix} a + 1 & b + 2 \\ 7 & -1 - c \end{pmatrix} \]

### Step 2: Set the resulting matrix equal to the right side
Now we set the resulting matrix equal to the matrix on the right side:
\[ \begin{pmatrix} a + 1 & b + 2 \\ 7 & -1 - c \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 7 & 3 \end{pmatrix} \]

### Step 3: Equate corresponding elements
From the equality of matrices, we can equate corresponding elements:

1. **From the first row, first column:**
\[ a + 1 = 5 \]

2. **From the first row, second column:**
\[ b + 2 = 0 \]

3. **From the second row, second column:**
\[ -1 - c = 3 \]

### Step 4: Solve for \(a\), \(b\), and \(c\)

1. **Solving for \(a\):**
\[ a + 1 = 5 \implies a = 5 - 1 = 4 \]

2. **Solving for \(b\):**
\[ b + 2 = 0 \implies b = 0 - 2 = -2 \]

3. **Solving for \(c\):**
\[ -1 - c = 3 \implies -c = 3 + 1 \implies -c = 4 \implies c = -4 \]

### Final Values
Thus, the values are:
- \(a = 4\)
- \(b = -2\)
- \(c = -4\)