Answer
The identity matrix can not be obtained on the left of the reduced row echelon form of $\left[A|I_{3}\right]$.
Thus, A has no inverse.
Work Step by Step
Procedure for Finding the Inverse of a Matrix
STEP 1: Form the matrix $\left[A|I_{n}\right]$
STEP 2: Transform the matrix $\left[A|I_{n}\right]$ into reduced row echelon form.
STEP 3: The reduced row echelon form of $\left[A|I_{n}\right]$ will contain the identity matrix $I_{n}$ on the left of the vertical bar;
the $n$ by $n$ matrix on the right of the vertical bar is the inverse of $A. $
If the identity matrix can not be obtained on the left, A has no inverse.
---
$\left[A|I_{3}\right]=\left[\begin{array}{rrr|rrr}
{1}&{1}&{-3}&{1}&{0}&{0}\\
{2}&{-4}&{1}&{0}&{1}&{0}\\
{-5}&{7}&{1}&{0}&{0}&{1}\end{array}\right]\rightarrow\left(\begin{array}{l}
.\\
R_{2}=-2r_{1}+r_{2}\\
R_{3}=5r_{1}+r_{3}
\end{array}\right)$
$\rightarrow\left[\begin{array}{rrr|rrr}
{1}&{1} &{-3} &{1} &{0}&{0}\\
{0}&{-6}&{7} &{-2}&{1}&{0}\\
{0}&{12} &{-14}&{5} &{0}&{1}\end{array}\right]\rightarrow\left(\begin{array}{l}
.\\
.\\
R_{3}=2r_{2}+r_{3}
\end{array}\right)$
$\rightarrow\left[\begin{array}{rrr|rrr}
{1}&{1} &{-3} &{1} &{0}&{0}\\
{0}&{-6}&{7} &{-2}&{1}&{0}\\
{0}&{ 0} &{0} &{1} & {2}&{1}\end{array}\right]$
The zeros on the left of the bar in the last row
make it impossible to obtain a leading nonzero entry in the third column.
The identity matrix can not be obtained on the left of the reduced row echelon form of $\left[A|I_{3}\right]$.
Thus, A has no inverse.