Solve System of Linear Equations Using Gauss-Jordan Elimination

Math Problem Statement

selesaikan dengan gauss jordan

To solve the system of linear equations using the Gauss-Jordan elimination method, let's first write down the system:

2x_1 - 4x_2 + x_3 + x_4 &= 0 \\ x_1 - 5x_2 + 2x_3 &= 0 \\ -2x_1 - 2x_2 - 2x_3 - x_4 &= 0 \\ x_1 + 3x_2 + x_4 &= 0 \\ x_1 - 2x_2 - x_3 + x_4 &= 0 \end{aligned}$$ We will now express this system as an augmented matrix and then apply the Gauss-Jordan elimination steps. ### Step 1: Form the Augmented Matrix The augmented matrix for this system is: $$\begin{pmatrix} 2 & -4 & 1 & 1 & | & 0 \\ 1 & -5 & 2 & 0 & | & 0 \\ -2 & -2 & -2 & -1 & | & 0 \\ 1 & 3 & 0 & 1 & | & 0 \\ 1 & -2 & -1 & 1 & | & 0 \end{pmatrix}$$ ### Step 2: Apply Gauss-Jordan Elimination **Row Operations:** 1. **R2 = R2 - 0.5 * R1** (to eliminate the $$x_1$$ term from $$R2$$): $$\begin{pmatrix} 2 & -4 & 1 & 1 & | & 0 \\ 0 & -3 & 1.5 & -0.5 & | & 0 \\ -2 & -2 & -2 & -1 & | & 0 \\ 1 & 3 & 0 & 1 & | & 0 \\ 1 & -2 & -1 & 1 & | & 0 \end{pmatrix}$$ 2. **R3 = R3 + R

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Linear Algebra
Gauss-Jordan Elimination

Advanced High School or College

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Solve System of Equations Using Gauss-Jordan Method: 2x + 2y - 2z = 22, 3x - y + z = -19, x - 4y + 2z = -74