To determine which matrix represents the given system of equations:
[tex]\[
\begin{aligned}
x - y &= 1 \\
2y + 3z &= -4 \\
x + y - 3z &= -1
\end{aligned}
\][/tex]
we need to convert the system into its augmented matrix form. An augmented matrix combines the coefficients of the variables on the left side with the constants on the right side of the equations.
1. Start with the first equation:
[tex]\[
x - y = 1
\][/tex]
The coefficients of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] are 1, -1, and 0 respectively, and the constant is 1. Therefore, the first row of the matrix is:
[tex]\[
[1, -1, 0, 1]
\][/tex]
2. Next, consider the second equation:
[tex]\[
2y + 3z = -4
\][/tex]
Here, the coefficients of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] are 0, 2, and 3 respectively, and the constant is -4. The second row of the matrix is:
[tex]\[
[0, 2, 3, -4]
\][/tex]
3. Finally, consider the third equation:
[tex]\[
x + y - 3z = -1
\][/tex]
The coefficients of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] are 1, 1, and -3 respectively, and the constant is -1. Thus, the third row of the matrix is:
[tex]\[
[1, 1, -3, -1]
\][/tex]
Combining all these rows, the augmented matrix for the system of equations is:
[tex]\[
\left[\begin{array}{ccc|c}
1 & -1 & 0 & 1 \\
0 & 2 & 3 & -4 \\
1 & 1 & -3 & -1
\end{array}\right]
\][/tex]
Looking at the given options, we can see that the correct matrix representation is:
C. [tex]\(\left[\begin{array}{ccc|c}1 & -1 & 0 & 1 \\ 0 & 2 & 3 & -4 \\ 1 & 1 & -3 & -1\end{array}\right]\)[/tex]
