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The matrix M is given as:

$$M = \begin{bmatrix} 7 & 4 \\ 5 & -3 \end{bmatrix}$$

For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is calculated as $ad - bc$.
For matrix M, we have $a=7$, $b=4$, $c=5$, and $d=-3$.

$$\det(M) = (7)(-3) - (4)(5)$$ $$\det(M) = -21 - 20$$ $$\det(M) = -41$$

Since the determinant of M is $-41$, which is not equal to zero ($\det(M) \neq 0$), the matrix M is invertible.

$$\boxed{\text{Since } \det(M) = -41 \neq 0, \text{ the matrix M is invertible.}}$$