Understand the additive inverse of a matrix.

The additive inverse of a matrix is found by negating each element of the matrix. If $A$ is a matrix, its additive inverse is $-A$, such that $A + (-A) = O$, where $O$ is the zero matrix.

The given matrix is $\begin{bmatrix} -1 & 2 \\ 3 & -4 \end{bmatrix}$.
Its additive inverse is:

We are given that the additive inverse is $\begin{bmatrix} a & -2 \\ -3 & b \end{bmatrix}$.
Therefore, we can set up the equality:

By comparing the elements in the same positions, we get:
$a = 1$
$b = 4$

The values are $a = 1$ and $b = 4$.

The final answer is $a=1, b=4$.