Find the value of the determinant. Show your work \( \left[\begin{array}{cc}12 & -7 \\ -4 & 3\end{array}\right] \)

To find the determinant of the given \( 2 \times 2 \) matrix: \[ \begin{pmatrix} 12 & -7 \\ -4 & 3 \\ \end{pmatrix} \] we can use the formula for the determinant of a \( 2 \times 2 \) matrix: \[ \text{Determinant} = ad - bc \] where the matrix is: \[ \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \] **Step-by-Step Calculation:** 1. **Identify the elements:** - \( a = 12 \) - \( b = -7 \) - \( c = -4 \) - \( d = 3 \) 2. **Apply the determinant formula:** \[ \text{Determinant} = (a \times d) - (b \times c) \] 3. **Substitute the values:** \[ \text{Determinant} = (12 \times 3) - (-7 \times -4) \] 4. **Perform the multiplications:** \[ 12 \times 3 = 36 \] \[ -7 \times -4 = 28 \] 5. **Subtract the second product from the first:** \[ \text{Determinant} = 36 - 28 = 8 \] **Final Answer:** The determinant of the matrix is **8**.