If \( \left[\begin{array}{cc}a+3 & 3 a-2 b \\ 3 a-c & a+b+c\end{array}\right]=\left[\begin{array}{cc}2 & -7+2 b \\ b+4 & 8 a\end{array}\right] \), evaluate \( a, b, c \)

To solve for \( a, b, c \), we start by setting the corresponding elements of the two matrices equal to each other. Given: \[ \left[\begin{array}{cc}a+3 & 3a-2b \\ 3a-c & a+b+c\end{array}\right] = \left[\begin{array}{cc}2 & -7+2b \\ b+4 & 8a\end{array}\right] \] We can write down the equations from the matrix equality: 1. \( a + 3 = 2 \) 2. \( 3a - 2b = -7 + 2b \) 3. \( 3a - c = b + 4 \) 4. \( a + b + c = 8a \) Now, we will solve these equations one by one. **From equation 1:** \[ a + 3 = 2 \implies a = 2 - 3 = -1 \] **Substituting \( a = -1 \) into equation 2:** \[ 3(-1) - 2b = -7 + 2b \\ -3 - 2b = -7 + 2b \\ -3 + 7 = 2b + 2b \\ 4 = 4b \\ b = \frac{4}{4} = 1 \] **Now substituting \( a = -1 \) and \( b = 1 \) into equation 3:** \[ 3(-1) - c = 1 + 4 \\ -3 - c = 5 \\ -c = 5 + 3 \\ -c = 8 \\ c = -8 \] **Finally, substituting \( a = -1, b = 1, c = -8 \) into equation 4 to verify:** \[ -1 + 1 + (-8) = 8(-1) \\ 0 - 8 = -8 \\ -8 = -8 \quad \text{(True)} \] Thus, the values of \( a, b, c \) are: \[ \boxed{-1}, \, \boxed{1}, \, \boxed{-8} \]