Let $A=\left[\begin{array}{ccc}1 & -2 & 3 \\ 1 & 2 & 1 \\ \lambda & 2 & -3\end{array}\right]$. If $A^{-1}$ does not exist, then $\lambda=$

-1

$A=\left[\begin{array}{ccc}1 & -2 & 3 \\ 1 & 2 & 1 \\ \lambda & 2 & -3\end{array}\right] $

$A^{-1}$ → doesn't exist

$\Rightarrow|A|=0$

so  $|A|=\left|\begin{array}{ccc}1 & -2 & 3 \\ 1 & 2 & 1 \\ \lambda & 2 & -3\end{array}\right|=0$

applying operations

$\left(\begin{array}{l}R_2 \rightarrow R_2+R_1 \\ R_3 \rightarrow R_3+R_1\end{array}\right)$

$|A|=\left|\begin{array}{ccc}1 & -2 & 3 \\ 2 & 0 & 4 \\ \lambda+1 & 0 & 0\end{array}\right|=0$

expanding along $R_3$

we get  $(\lambda+1)\left|\begin{array}{cc}
-2 & 3 \\ 0 & 4 \end{array}\right|+0+0=0$

$= (\lambda+1)(-8)=0$

so  $\lambda=-1$