Find the rank of the following matrix depending on $\lambda\in\Bbb R$. $$A=\begin{pmatrix} 1&2&3&4\\ 2&\lambda&6&7\\ 3&6&8&9\\ 4&7&9&10 \end{pmatrix}$$
My attempt:
$$\begin{pmatrix} 1&2&3&4\\ 2&\lambda&6&7\\ 3&6&8&9\\ 4&7&9&10 \end{pmatrix}\sim\begin{pmatrix} 1&2&3&4\\ 0&\lambda-4&0&-1\\ 0&0&-1&-3\\ 0&-1&-3&-6 \end{pmatrix}\sim\begin{pmatrix} 1&0&-3&-8\\ 0&\lambda-4&0&-1\\ 0&0&-1&-3\\ 0&-1&-3&-6\\ \end{pmatrix}$$ For $\lambda=4$ we have: $$\begin{pmatrix} 1 &0&-3&-8\\ 0&0&0&-1\\ 0&0&-1&-3\\ 0&-1&-3&-6 \end{pmatrix}\sim\begin{pmatrix} 1&0&0&1\\ 0&0&0&-1\\ 0&0&-1&-3\\ 0&-1&0&3\end{pmatrix}\sim\begin{pmatrix} 1&0&0&0\\ 0&0&0&-1\\ 0&0&-1&0\\ 0&-1&0&0\\ \end{pmatrix}$$
$\Rightarrow r(A)=4$
For $\lambda\neq 4$ we have:
$$\begin{pmatrix} 1&0&0&1\\ 0&\lambda-4&0&-1\\ 0&0&-1&-3\\ 0&-1&0&3\\ \end{pmatrix}\sim\begin{pmatrix} 1&0&0&1\\ 0&0&0&3\lambda-13\\ 0&0&-1&-3\\ 0&-1&0&3 \end{pmatrix}$$
For$\lambda=\frac{13}{3}\Rightarrow r(A)=3$ and for $\lambda\neq \frac{13}{3} \Rightarrow r(A)=4$
Is this correct? Thanks!