Answer
The determinant for the given matrix is zero, so the matrix $A$ has no inverse.
Work Step by Step
The determinant for the $3 \times 3$ matrix can be obtained as:
$det=\begin{vmatrix} a & b & c \\ d &e & f \\g &h & i \\
\end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)$
Recall the determinant property that if $D=0$, then the matrix has no inverse.
We are given
$A=\begin{vmatrix} 1 & 1 & -3 \\ 2 &-4 & 1 \\-5 &7 & 1 \\ \end{vmatrix}$
Now,
$det A=1((−4)−1⋅7)−1(2−(−5))+(−3)(2(7)−(−4)(−5))=(1)(−11)−(1)(7)+(−3)(−6)\\=−11−7+18\\=0$
We can conclude that the determinant for the given matrix is zero and thus the matrix $A$ has no inverse.
