To solve the system of linear equations by the matrix inversion method, we represent the system in the form [tex]\(AX = B\)[/tex], where [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(X\)[/tex] is the column vector of variables, and [tex]\(B\)[/tex] is the constant column vector.
Let's start with part (e):
### Part (e)
Given system of equations:
[tex]\[
\begin{cases}
x + 2y + z = 7 \\
2x - y + z = 3 \\
3x + y + 2z = 8
\end{cases}
\][/tex]
First, write this system in matrix form:
[tex]\[
A = \begin{pmatrix}
1 & 2 & 1 \\
2 & -1 & 1 \\
3 & 1 & 2 \\
\end{pmatrix}, \quad
X = \begin{pmatrix}
x \\
y \\
z \\
\end{pmatrix}, \quad
B = \begin{pmatrix}
7 \\
3 \\
8 \\
\end{pmatrix}
\][/tex]
We need to find the inverse of matrix [tex]\(A\)[/tex]. We'll denote this inverse by [tex]\(A^{-1}\)[/tex]. Once we have [tex]\(A^{-1}\)[/tex], we can find the solution [tex]\(X\)[/tex] using:
[tex]\[
X = A^{-1}B
\][/tex]
Let's compute [tex]\(A^{-1}\)[/tex]:
1. Compute the determinant of [tex]\(A\)[/tex]:
[tex]\[
\text{det}(A) = 1 \cdot
\begin{vmatrix}
-1 & 1 \\
1 & 2
\end{vmatrix}
- 2 \cdot
\begin{vmatrix}
2 & 1 \\
3 & 2
\end{vmatrix}
+ 1 \cdot
\begin{vmatrix}
2 & -1 \\
3 & 1
\end{vmatrix}
\][/tex]
[tex]\[
\text{det}(A) = 1 \cdot ((-1) \cdot 2 - 1 \cdot 1) - 2 \cdot (2 \cdot 2 - 1 \cdot 3) + 1 \cdot (2 \cdot 1 - (-1) \cdot 3)
\][/tex]
[tex]\[
\text{det}(A) = 1 \cdot (-2 - 1) - 2 \cdot (4 - 3) + 1 \cdot (2 + 3)
\][/tex]
[tex]\[
\text{det}(A) = -3 - 2 + 5 = 0
\][/tex]
Since the determinant of [tex]\(A\)[/tex] is 0, the matrix [tex]\(A\)[/tex] is not invertible, and thus we cannot use the matrix inversion method to solve this system of equations.
Given that part (e) cannot be solved using matrix inversion due to the determinant being zero, it indicates that we need to check for linear dependency in the equations or consider an alternate method like Gaussian elimination.
For detailed processing for other parts (a) and (b) and (d), ensure to set up the equations correctly and confirm if the matrix is invertible before proceeding with the matrix inversion method, employing similar steps to those shown above.
