Consider the following matrix. \[ A=\left[\begin{array}{ll} -3 & 4 \\ -2 & 3 \end{array}\right] \] Choose the correct description of $A$. Find $A^{-1}$ if it exists. - $A$ is nonsingular. That is, it has an inverse. \[ A^{-1}= \] $A$ is singular. That is; its inverse doesn't exist.

Step 1 ：Given the matrix A = \(\begin{bmatrix} -3 & 4 \\ -2 & 3 \end{bmatrix}\)

Step 2 ：First, we need to calculate the determinant of the matrix A. The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is given by \(ad - bc\).

Step 3 ：Substituting the values from matrix A, we get \((-3)(3) - (4)(-2) = -9 + 8 = -1\).

Step 4 ：Since the determinant of the matrix A is not zero, the matrix is nonsingular and has an inverse.

Step 5 ：The formula to find the inverse of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is \(\frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\).

Step 6 ：Substituting the values from matrix A and its determinant into the formula, we get \(\frac{1}{-1} \begin{bmatrix} 3 & -4 \\ 2 & -3 \end{bmatrix} = \begin{bmatrix} -3 & 4 \\ -2 & 3 \end{bmatrix}\).

Step 7 ：\(\boxed{A^{-1} = \begin{bmatrix} -3 & 4 \\ -2 & 3 \end{bmatrix}}\)