Find the inverse of the following matrix, if possible. Reduce all fractions to lowest terms. \[ A=\left[\begin{array}{ccc}-2 & 0 & -3 \\ 3 & 4 & 3 \\ 1 & 1 & -2\end{array}\right] \] Answer Selecting an option will display any text boxes needed to complete your answer. Does this matrix have an inverse?

To determine if the matrix \( A \) has an inverse, we need to calculate its determinant. A matrix has an inverse if and only if its determinant is non-zero. Let's calculate the determinant of \( A \): \[ A = \begin{bmatrix} -2 & 0 & -3 \\ 3 & 4 & 3 \\ 1 & 1 & -2 \end{bmatrix} \] The determinant can be calculated using the formula for a \( 3 \times 3 \) matrix: \[ \text{det}(A) = a(ei-fh) - b(dg-eh) + c(dh-eg) \] where \( a, b, c \) are the elements of the first row, and \( d, e, f, g, h, i \) are the other elements as follows: \[ \text{det}(A) = -2 \left(4 \cdot (-2) - 3 \cdot 1\right) - 0 + (-3)\left(3 \cdot 1 - 4 \cdot 1\right) \] \[ = -2 \left(-8 - 3\right) - 3 \cdot (-1) \] \[ = -2 \cdot (-11) + 3 \] \[ = 22 + 3 = 25 \] Since the determinant is 25 (not zero), the matrix \( A \) does have an inverse. To find the inverse, we can use the formula \( A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \), where \( \text{adj}(A) \) is the adjugate of \( A \). The adjugate can be determined by finding the matrix of cofactors and then transposing that matrix. Now that we know \( A \) is invertible, we can systematically calculate its inverse!