To determine which of the given matrices has an inverse, we need to check the criteria for invertibility. Specifically, a matrix is invertible if it is a square matrix (i.e., it has the same number of rows and columns) and its determinant is non-zero.
Let's analyze each matrix individually:
1. [tex]\(\left[\begin{array}{c}-6 \\ 3\end{array}\right]\)[/tex]
- This is a [tex]\(2 \times 1\)[/tex] matrix (2 rows and 1 column).
- Only square matrices can have an inverse.
- Therefore, this matrix does not have an inverse.
2. [tex]\(\left[\begin{array}{lll}4 & -2 & 1\end{array}\right]\)[/tex]
- This is a [tex]\(1 \times 3\)[/tex] matrix (1 row and 3 columns).
- Again, only square matrices can have an inverse.
- Therefore, this matrix does not have an inverse.
3. [tex]\(\left[\begin{array}{cc}6 & -9 \\ 2 & 1\end{array}\right]\)[/tex]
- This is a [tex]\(2 \times 2\)[/tex] matrix (2 rows and 2 columns), so it is a square matrix.
- We need to check if its determinant is non-zero.
The determinant of a [tex]\(2 \times 2\)[/tex] matrix [tex]\(\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]\)[/tex] is calculated as [tex]\(ad - bc\)[/tex]:
[tex]\[
\begin{vmatrix} 6 & -9 \\ 2 & 1 \end{vmatrix}
= (6 \cdot 1) - (-9 \cdot 2)
= 6 + 18
= 24
\][/tex]
- Since the determinant is 24 (non-zero), this matrix has an inverse.
4. [tex]\(\left[\begin{array}{ccc}-3 & 5 & 1 \\ 4 & -6 & 0\end{array}\right]\)[/tex]
- This is a [tex]\(2 \times 3\)[/tex] matrix (2 rows and 3 columns).
- As it is not a square matrix, it cannot have an inverse.
Based on the analysis above, the only matrix that has an inverse is the third one:
[tex]\[
\left[\begin{array}{cc}6 & -9 \\ 2 & 1\end{array}\right]
\][/tex]
