To solve this system of equations:
[tex]$
\begin{aligned}
x - y - z &= 4 \\
-x + 2y + 3z &= -2 \\
x + y + 4z &= 10
\end{aligned}
$[/tex]
we can express the system in matrix form where [tex]\(AX = B\)[/tex]. Here, [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(X\)[/tex] is the matrix of variables, and [tex]\(B\)[/tex] is the constants matrix. The matrices are as follows:
[tex]$
A = \begin{bmatrix} 1 & -1 & -1 \\ -1 & 2 & 3 \\ 1 & 1 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 4 \\ -2 \\ 10 \end{bmatrix}
$[/tex]
To find the solution for [tex]\(X\)[/tex], we need to find the inverse of matrix [tex]\(A\)[/tex], denoted as [tex]\(A^{-1}\)[/tex]. Once we have [tex]\(A^{-1}\)[/tex], we can compute [tex]\(X\)[/tex] using the formula:
[tex]$
X = A^{-1} B
$[/tex]
The matrix [tex]\(A^{-1}\)[/tex] is given as:
[tex]$
A^{-1} = \begin{bmatrix} 5 & 3 & -1 \\ 7 & 5 & -2 \\ -3 & -2 & 1 \end{bmatrix}
$[/tex]
To identify the correct inverse matrix from the options provided, we match this result with the given choices:
A. [tex]\(\begin{bmatrix} -5 & 3 & -1 \\ -7 & 5 & -2 \\ -3 & 2 & 1 \end{bmatrix}\)[/tex]
B. [tex]\(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)[/tex]
C. [tex]\(\begin{bmatrix} 5 & 3 & -1 \\ 7 & 5 & -2 \\ -3 & -2 & 1 \end{bmatrix}\)[/tex]
D. [tex]\(\begin{bmatrix} 5 & 2 & 1 \\ 7 & 5 & 2 \\ 5 & 3 & -1 \end{bmatrix}\)[/tex]
The correct answer, based on the calculated inverse matrix, is:
C. [tex]\(\begin{bmatrix} 5 & 3 & -1 \\ 7 & 5 & -2 \\ -3 & -2 & 1 \end{bmatrix}\)[/tex]
