Question

Use Gauss-Jordan row reduction to solve the given system of equations. HINT [See Examples 1-6.] (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer in terms of x, where y = y(x) and z = z(x).) − 1 2 x + y − 1 2 z = 0 − 1 2 x − 1 2 y + z = 0 x − 1 2 y − 1 2 z = 0

Answer 1

Answer:

$x = 0, y = 0, z= 0$

Step-by-step explanation:

The idea of Gauss Jordan row reduction is to reduce the matrix form of the system of equation to reduced row echelon form using some elementary row operations. There are three operations that can be performed on the rows, namely;

1. swap two rows.

2. multiply one row by a nonzero number

3. add or substract the multiple of one row to/from another

For the given exercise, the matrix form is given as

$\left[\begin{array}{cccc} -12&1&-12 & 0\\-12&-12&1 & 0\\1&-12&-12&0\end{array}\right]$

These are the operations that are performed to get the reduced row echelon form.

1. swap Row1 and Row3

$\left[\begin{array}{cccc} 1&-12&-12 & 0\\-12&-12&1 & 0\\-12&1&-12&0\end{array}\right]$

2. Row2 + 12*Row1 and Row3 + 12*Row1

$\left[\begin{array}{cccc} 1&-12&-12 & 0\\0&-156&-143 & 0\\0&-143&-156&0\end{array}\right]$

3. -1/156 * Row2

$\left[\begin{array}{cccc} 1&-12&-12 & 0\\0&1&143/156 & 0\\0&-143&-156&0\end{array}\right]$

4. Row1 + 12*Row2 and Row3 + 143*Row2

$\left[\begin{array}{cccc} 1&0&-1 & 0\\0&1&143/156 & 0\\0&0&-3837/156&0\end{array}\right]$

5. -156/3837 * Row3

$\left[\begin{array}{cccc} 1&0&-1 & 0\\0&1&143/156 & 0\\0&0&1&0\end{array}\right]$

6. Row1 + Row3

$\left[\begin{array}{cccc} 1&0&0 & 0\\0&1&143/156 & 0\\0&0&1&0\end{array}\right]$

The matrix is now reduced to the row echelon form.

From Row1, the element in the first column is equal to 1 and the last column is equal to zero, this implies that $x = 0$.

From Row2, the element in the second column is equal to 1, the third column is equal to 143/156 and last column is zero, this means that $y + 143/156 z = 0$

Similarly from Row3, $z = 0$.

Putting these together, $x = 0, y = 0, z= 0$