If `A=[{:(,3,a),(,-4,8):}], B=[{:(,c,4),(,-3,0):}] , C=[{:(,-1,4),(,3,b):}]` and 3A-2C=6B. Find the values of a,b and c.

Answer

To solve the equation \(3A - 2C = 6B\) given the matrices \(A\), \(B\), and \(C\), we will follow these steps:

### Step 1: Write down the matrices
We have:
\[ A = \begin{pmatrix} 3 & a \\ -4 & 8 \end{pmatrix}, \quad B = \begin{pmatrix} c & 4 \\ -3 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} -1 & 4 \\ 3 & b \end{pmatrix} \]

### Step 2: Calculate \(3A\)
To find \(3A\), we multiply each element of matrix \(A\) by 3:
\[ 3A = 3 \cdot \begin{pmatrix} 3 & a \\ -4 & 8 \end{pmatrix} = \begin{pmatrix} 9 & 3a \\ -12 & 24 \end{pmatrix} \]

### Step 3: Calculate \(2C\)
Next, we find \(2C\) by multiplying each element of matrix \(C\) by 2:
\[ 2C = 2 \cdot \begin{pmatrix} -1 & 4 \\ 3 & b \end{pmatrix} = \begin{pmatrix} -2 & 8 \\ 6 & 2b \end{pmatrix} \]

### Step 4: Set up the equation \(3A - 2C\)
Now, we can substitute \(3A\) and \(2C\) into the equation:
\[ 3A - 2C = \begin{pmatrix} 9 & 3a \\ -12 & 24 \end{pmatrix} - \begin{pmatrix} -2 & 8 \\ 6 & 2b \end{pmatrix} = \begin{pmatrix} 9 + 2 & 3a - 8 \\ -12 - 6 & 24 - 2b \end{pmatrix} \]
This simplifies to:
\[ 3A - 2C = \begin{pmatrix} 11 & 3a - 8 \\ -18 & 24 - 2b \end{pmatrix} \]

### Step 5: Calculate \(6B\)
Next, we calculate \(6B\) by multiplying each element of matrix \(B\) by 6:
\[ 6B = 6 \cdot \begin{pmatrix} c & 4 \\ -3 & 0 \end{pmatrix} = \begin{pmatrix} 6c & 24 \\ -18 & 0 \end{pmatrix} \]

### Step 6: Set the two sides equal
Now we equate \(3A - 2C\) and \(6B\):
\[ \begin{pmatrix} 11 & 3a - 8 \\ -18 & 24 - 2b \end{pmatrix} = \begin{pmatrix} 6c & 24 \\ -18 & 0 \end{pmatrix} \]

### Step 7: Set up equations from the matrix equality
From the matrix equality, we can derive the following equations:
1. \(11 = 6c\)
2. \(3a - 8 = 24\)
3. \(-18 = -18\) (This equation is always true)
4. \(24 - 2b = 0\)

### Step 8: Solve for \(c\)
From the first equation:
\[ 6c = 11 \implies c = \frac{11}{6} \]

### Step 9: Solve for \(a\)
From the second equation:
\[ 3a - 8 = 24 \implies 3a = 24 + 8 = 32 \implies a = \frac{32}{3} \]

### Step 10: Solve for \(b\)
From the fourth equation:
\[ 24 - 2b = 0 \implies 2b = 24 \implies b = 12 \]

### Final Values
Thus, the values are:
\[ a = \frac{32}{3}, \quad b = 12, \quad c = \frac{11}{6} \]