The asymptotic properties of estimators conditional on a fixed estimator

Question

Suppose we have:

1. a 2 parameter model $(\mu,\sigma)$
2. Maximum likelihood estimators $(\hat{\lambda},\hat{\kappa})$
3. score function $sc(\lambda,\kappa) = [sc_1(\lambda,\kappa),sc_2(\lambda,\kappa)]$
4. Expected information matrix evaluated at the true parameters $n \begin{bmatrix} a & b\\ c & d\end{bmatrix}$

Suppose

$n^{\frac{1}{2}} \begin{bmatrix} \hat{\lambda} &&- \lambda_0 \\ \hat{\kappa} && -\kappa_0 \end{bmatrix} \approx \begin{bmatrix} a & b\\ c & d\end{bmatrix} ^{-1} \begin{bmatrix} s_1 \\s_2\end{bmatrix}$

as $n\rightarrow\infty$ where $s_1 = n^{-\frac{1}{2}}sc_1(\lambda_0,\kappa_0) $ and $s_2 = n^{-\frac{1}{2}}sc_2(\lambda_0,\kappa_0)$

Then how can we show that $n^{\frac{1}{2}} \{\hat{\lambda}(\kappa)-\lambda_0\} \approx -\frac{1}{a}s_1$