Simplify the Matrix

Problem

[[x,−3*y,14*z,7],[−2*x,8*y,−38*z,−10],[x,−2*y,9*z,11]]

Solution

1. Identify the matrix as a system of linear equations or a matrix to be put into row-echelon form. We will use Gaussian elimination to simplify the rows.

2. Eliminate the x terms in the second and third rows by using the first row as a pivot.

3. Perform the row operation (R_2)→(R_2)+2*(R_1)

(R_2)→[[−2*x+2*(x),8*y+2*(−3*y),−38*z+2*(14*z),−10+2*(7)]]

(R_2)→[[0,2*y,−10*z,4]]

4. Perform the row operation (R_3)→(R_3)−(R_1)

(R_3)→[[x−x,−2*y−(−3*y),9*z−14*z,11−7]]

(R_3)→[[0,y,−5*z,4]]

5. Update the matrix with the new rows.

[[x,−3*y,14*z,7],[0,2*y,−10*z,4],[0,y,−5*z,4]]

6. Eliminate the y term in the third row by using the second row. Perform (R_3)→(R_3)−1/2*(R_2)

(R_3)→[[0,y−1/2*(2*y),−5*z−1/2*(−10*z),4−1/2*(4)]]

(R_3)→[[0,0,0,2]]

7. Analyze the resulting matrix. The third row 0 = 2$ indicates that if this matrix represents a system of linear equations, the system is inconsistent and has no solution.

Final Answer

[[x,−3*y,14*z,7],[−2*x,8*y,−38*z,−10],[x,−2*y,9*z,11]]⇒[[x,−3*y,14*z,7],[0,2*y,−10*z,4],[0,0,0,2]]

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