How many pivots does the reduced row echelon form of the following linear system have? (We pondered this linear system earlier. You should be able to see this at a glance. No calculations necessary!) $$\begin{eqnarray} x + 2y +3z &=& 4\\ 2x + 4y +6z &=& 8\\ -x - 2y -3z &=& -4\\ \end{eqnarray}$$简答题
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Question

How many pivots does the reduced row echelon form of the following linear system have? (We pondered this linear system earlier. You should be able to see this at a glance. No calculations necessary!) $$\begin{eqnarray} x + 2y +3z &=& 4\\ 2x + 4y +6z &=& 8\\ -x - 2y -3z &=& -4\\ \end{eqnarray}$$简答题

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BlackTom AI · Accepted Answer

The system presented consists of three equations:
- x + 2y + 3z = 4
- 2x + 4y + 6z = 8
- -x - 2y - 3z = -4

A quick glance shows that the second equation is exactly twice the first, and the third is the negative of the first. This means all three equations represent the same plane, i.e., there is only one independent equation among the three.

In reduced row echelon form, the number of pivots equals the rank of the coefficient matrix, which corresponds to the number of linearly independent equations. Since there is only one independent equation, the coefficient matrix has rank 1, so there is exactly one pivot.

To visualize, after row-reducing, you would obtain a leading 1 in the first column of the first row and zeros elsewhere in that column, with the remaining rows being zero rows. Thus the RREF has a single pivot overall.