Simplify the Matrix [[x,y,z,6],[x,-y,z,7]]

Problem

[[x,y,z,6],[x,−y,z,7]]

Solution

1. Identify the goal to simplify the matrix using Gaussian elimination to reach row-echelon form.

2. Subtract the first row from the second row ((R_2)−(R_1)→(R_2) to eliminate the x and z terms in the second row.

[[x,y,z,6],[x−x,−y−y,z−z,7−6]]

3. Simplify the arithmetic in the second row.

[[x,y,z,6],[0,−2*y,0,1]]

4. Divide the second row by −2 (−1/2*(R_2)→(R_2) to normalize the y coefficient.

[[x,y,z,6],[0,y,0,−1/2]]

5. Subtract the new second row from the first row ((R_1)−(R_2)→(R_1) to eliminate the y term in the first row.

[[x,y−y,z,6−(−1/2)],[0,y,0,−1/2]]

6. Simplify the constants in the first row.

[[x,0,z,13/2],[0,y,0,−1/2]]

Final Answer

[[x,y,z,6],[x,−y,z,7]]=[[x,0,z,13/2],[0,y,0,−1/2]]

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