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!) [math: x+2y+3z=42x+4y+6z=8−x−2y−3z=−4]\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!) [math: x+2y+3z=42x+4y+6z=8−x−2y−3z=−4]\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

We are given a system with three equations: x + 2y + 3z = 4, 2x + 4y + 6z = 8, and -x - 2y - 3z = -4.

First, observe that each of the second and third equations is a constant multiple of the first equation: the second is 2 times the first, and the third is -1 times the first. This means all three equations describe the same plane in (x, y, z) space, so there is only one independent equation among them.

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 rank is 1, and therefore there is exactly one pivot.

Intuitively, with three variables but only one independent constraint, you can have infinitely many solutions along the directions that keep that single constraint satisfied, but the question is specifically about how many pivots appear in the RREF, which is 1 in this case.

Thus, the reduced row echelon form would contain a single leading 1 (pivot) in the first column, with the remaining rows being zero rows, reflecting the dependence among the original equations.

类似问题

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!) $$\begin{eqnarray} x + 2y +3z &=& 4\\ 2x + 4y +6z &=& 8\\ -x - 2y -3z &=& -4\\ \end{eqnarray}$$

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