Answer
$x_1=\frac{\det(B_1)}{\det(A)}=0$
$x_2=\frac{\det(B_2)}{\det(A)}=0$
$x_3=\frac{\det(B_3)}{\det(A)}=0$
Work Step by Step
We are given:
$x_1-3x_2+x_3=0$
$x_1+4x_2-x_3=0$
$2x_1+x_2-3x_3=0$
which can also be written as:
$A=\begin{bmatrix}
1 & -3&1\\
1 & 4&-1\\
2&1&-3
\end{bmatrix} \rightarrow \det(A)=-21$
$B_1=\begin{bmatrix}
0 &-3&1\\
0 & 4&-1\\
0&1&-3
\end{bmatrix} \rightarrow \det(A)=0$
$B_2=\begin{bmatrix}
1 &0&1\\
1& 0&-1\\
2&0&-3
\end{bmatrix} \rightarrow \det(A)=0$
$B_3=\begin{bmatrix}
1&-3&0\\
1& 4&0\\
2&1&0
\end{bmatrix} \rightarrow \det(A)=0$
Use Cramer’s rule: $x_k=\frac{\det(B_k)}{\det(A)}$ to find the results:
$x_1=\frac{\det(B_1)}{\det(A)}=0$
$x_2=\frac{\det(B_2)}{\det(A)}=0$
$x_3=\frac{\det(B_3)}{\det(A)}=0$
