The $SL_{2}(\mathbb{Z})$ double coset of diagonal matrix

Question

I have a trouble proving that: For $k\in \mathbb{N}$, the double coset \begin{align*} SL_{2}(\mathbb{Z})\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} SL_{2}(\mathbb{Z})\end{align*} is consisting of the matrices \begin{pmatrix} a & b \\ c & d \end{pmatrix} with integer entries having gcd 1 and of determinant $k$. I already know that the smith normal form of $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ gives a matrix $\gamma_{1}, \gamma_{2}\in GL_{2}(\mathbb{Z})$ such that $\gamma_{1} \,\text{diag}(k,1)\,\gamma_{2}$ is equal to \begin{pmatrix} a & b \\ c & d \end{pmatrix}. How can i make $\gamma_{1},\gamma_{2}$ are contained in $SL_{2}(\mathbb{Z})$? Or is there any other way to prove the statement above?