To determine which matrix does not have an inverse, we need to check the determinant of each matrix. If the determinant of a matrix is zero, then the matrix does not have an inverse.
1. Matrix A:
[tex]\[
A = \left[\begin{array}{cc}1 & 3 \\ -4 & 9\end{array}\right]
\][/tex]
The determinant of [tex]\(A\)[/tex] is calculated as:
[tex]\[
\text{det}(A) = (1 \cdot 9) - (3 \cdot -4) = 9 + 12 = 21
\][/tex]
Since the determinant is not zero, matrix [tex]\(A\)[/tex] has an inverse.
2. Matrix B:
[tex]\[
B = \left[\begin{array}{cc}-6 & -4 \\ 3 & -2\end{array}\right]
\][/tex]
The determinant of [tex]\(B\)[/tex] is calculated as:
[tex]\[
\text{det}(B) = (-6 \cdot -2) - (-4 \cdot 3) = 12 + 12 = 24
\][/tex]
Since the determinant is not zero, matrix [tex]\(B\)[/tex] has an inverse.
3. Matrix C:
[tex]\[
C = \left[\begin{array}{cc}6 & 2 \\ 3 & 0\end{array}\right]
\][/tex]
The determinant of [tex]\(C\)[/tex] is calculated as:
[tex]\[
\text{det}(C) = (6 \cdot 0) - (2 \cdot 3) = 0 - 6 = -6
\][/tex]
Since the determinant is not zero, matrix [tex]\(C\)[/tex] has an inverse.
4. Matrix D:
[tex]\[
D = \left[\begin{array}{cc}-4 & -2 \\ 8 & 4\end{array}\right]
\][/tex]
The determinant of [tex]\(D\)[/tex] is calculated as:
[tex]\[
\text{det}(D) = (-4 \cdot 4) - (-2 \cdot 8) = -16 + 16 = 0
\][/tex]
Since the determinant is zero, matrix [tex]\(D\)[/tex] does not have an inverse.
Therefore, the matrix that does not have an inverse is:
[tex]\[
\boxed{D} \left[\begin{array}{cc}-4 & -2 \\ 8 & 4\end{array}\right]
\][/tex]
