Given the matrix:
[tex]\[
\begin{pmatrix}
-4 & -6 \\
2 & 3
\end{pmatrix}
\][/tex]
We are seeking to determine its inverse and compare it with the given options. To find the inverse of a [tex]\(2 \times 2\)[/tex] matrix [tex]\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)[/tex], the inverse (if it exists) is given by:
[tex]\[
\frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\][/tex]
provided that [tex]\(ad - bc \neq 0\)[/tex]. For our matrix, [tex]\(a = -4\)[/tex], [tex]\(b = -6\)[/tex], [tex]\(c = 2\)[/tex], and [tex]\(d = 3\)[/tex].
First, we compute the determinant:
[tex]\[
ad - bc = (-4)(3) - (-6)(2) = -12 + 12 = 0
\][/tex]
Because the determinant is zero, the matrix does not have an inverse. Consequently, among the given options, the correct response must be that the inverse does not exist.
Answer: [tex]\(\boxed{D \text{. Does Not Exist}}\)[/tex]
