The problem asks for the inverse of the matrix $M = \begin{pmatrix} 4 & 3 \\ -3 & -2 \end{pmatrix}$.

To find the inverse of a $2 \times 2$ matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the formula is:

$M^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

Let's apply this formula:

1. Identify the elements:

   • $a = 4$
   • $b = 3$
   • $c = -3$
   • $d = -2$

2. Calculate the determinant $ad - bc$: $\text{Determinant} = (4)(-2) - (3)(-3) = -8 + 9 = 1$

3. Apply the formula for the inverse: $M^{-1} = \frac{1}{1} \begin{pmatrix} -2 & -3 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} -2 & -3 \\ 3 & 4 \end{pmatrix}$

So, the inverse of the matrix $M$ is:

$M^{-1} = \begin{pmatrix} -2 & -3 \\ 3 & 4 \end{pmatrix}$

Would you like further details or have any questions?

Here are some related questions:

1. How do you find the determinant of a $3 \times 3$ matrix?
2. What happens if the determinant of a matrix is zero?
3. How can the inverse of a matrix be used to solve systems of equations?
4. What are some practical applications of matrix inverses in real life?
5. How do you compute the inverse of larger matrices using row reduction?

Tip: Always check the determinant before calculating the inverse of a matrix; if the determinant is zero, the matrix does not have an inverse.