To determine which matrix does not have an inverse, we need to calculate the determinant of each matrix. A matrix is invertible if and only if its determinant is non-zero. Let's examine each matrix:
1. Matrix [tex]\( A \)[/tex]:
[tex]\[
A = \begin{pmatrix} -6 & -4 \\ 3 & -2 \end{pmatrix}
\][/tex]
2. Matrix [tex]\( B \)[/tex]:
[tex]\[
B = \begin{pmatrix} 6 & 2 \\ 3 & 0 \end{pmatrix}
\][/tex]
3. Matrix [tex]\( C \)[/tex]:
[tex]\[
C = \begin{pmatrix} 1 & 3 \\ -4 & 9 \end{pmatrix}
\][/tex]
4. Matrix [tex]\( D \)[/tex]:
[tex]\[
D = \begin{pmatrix} -4 & -2 \\ 8 & 4 \end{pmatrix}
\][/tex]
Step-by-step calculations for determinants:
1. Determinant of Matrix [tex]\( A \)[/tex]:
[tex]\[
\det(A) = (-6 \times -2) - (-4 \times 3) = 12 - (-12) = 12 + 12 = 24
\][/tex]
2. Determinant of Matrix [tex]\( B \)[/tex]:
[tex]\[
\det(B) = (6 \times 0) - (2 \times 3) = 0 - 6 = -6
\][/tex]
3. Determinant of Matrix [tex]\( C \)[/tex]:
[tex]\[
\det(C) = (1 \times 9) - (3 \times -4) = 9 - (-12) = 9 + 12 = 21
\][/tex]
4. Determinant of Matrix [tex]\( D \)[/tex]:
[tex]\[
\det(D) = (-4 \times 4) - (-2 \times 8) = -16 - (-16) = -16 + 16 = 0
\][/tex]
Interpretation:
- The determinant of Matrix [tex]\( A \)[/tex] is 24.
- The determinant of Matrix [tex]\( B \)[/tex] is -6.
- The determinant of Matrix [tex]\( C \)[/tex] is 21.
- The determinant of Matrix [tex]\( D \)[/tex] is 0.
Since a matrix is not invertible if its determinant is zero, we can see that Matrix [tex]\( D \)[/tex] does not have an inverse because its determinant is 0.
Therefore, the matrix that does not have an inverse is:
[tex]\[ \boxed{D} \][/tex]
