Using matrix algebra for systems of linear equations in more than two variables
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Consider the following system of linear equations:
\[
\begin{aligned}
x + y + z &= 1 \\
4x - 3y + 4z &= 32 \\
x - 10y - 2z &= 27
\end{aligned}
\]
i) Write the system in the matrix form as \( A X = K \)
ii) Find \( A^{-1} \)
iii) Hence solve the system of equations.
i) \(\left[\begin{array}{ccc}
1 & 1 & 1 \\
4 & -3 & 4 \\
1 & -10 & -2
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{c}
1 \\
32 \\
27
\end{array}\right]\)
ii) \(\dfrac{1}{21}\left[\begin{array}{ccc}46 & -8 & 7 \\ 12 & -3 & 0 \\ -37 & 11 & -7\end{array}\right]\)
iii) \(x =-1, y=-4, z=6\)
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