Turnbull George
07/19/2024 · Elementary School
5) Consider the following matrix: \[ \left[\begin{array}{rrrrrr}1 & 0 & 0 & 2 & \vdots & 2 \\ 0 & 1 & 4 & 5 & \vdots & 0 \\ 0 & 0 & 1 & -1 & \vdots & -1 \\ 0 & 0 & 0 & 1 & \vdots & 2\end{array}\right] \] (i) Write down a system of linear equations which has this augmented matrix. (ii) Use back substitution to determine all solutions of this system.
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The system of linear equations corresponds to:
1. \( x_1 + 2x_4 = 2 \)
2. \( x_2 + 4x_3 + 5x_4 = 0 \)
3. \( x_3 - x_4 = -1 \)
4. \( x_4 = 2 \)
Using back substitution, the solutions are:
\[
x_1 = -2, \quad x_2 = -14, \quad x_3 = 1, \quad x_4 = 2
\]
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