Solving systems: elimination, row echelon and rank
Rewrite a system without changing its solutions, build a staircase of pivots, and read both the answer and the rank from what remains.
The previous systems lesson asked whether the clues pin an answer down. This one supplies the algorithm: keep the constants attached, make one legal row move at a time, and clear one pivot column after another.
By the end, row echelon form is not a display format. It is the picture that exposes rank, free variables, contradictions, and the values recovered by back-substitution.
Every move must preserve the system. Work one pivot at a time; matrices, solutions and verdicts are computed exactly, so an illegal move cannot become correct just because its numbers look simpler.
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Change the clues, not their shared answer
Elimination rewrites a system into an easier equivalent system.
Solving by elimination does not guess values. It replaces the equations with new equations that have exactly the same simultaneous solutions, until one variable is isolated.
If two equations are both true, their difference is true too. Subtracting one row from another can make a coefficient zero, removing a variable from that equation without removing any valid solution.
From x + y = 7 and x - y = 1, subtracting the second from the first gives 2y = 6. The easier equation carries information the original pair already contained.
Challenge
Preserve the Answer
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Choose the rewrite that keeps exactly the same solution set.
Simpler is useful only when it is equivalent. Each correct move can be reversed.
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Only three row operations are legal
Swap rows, scale by a nonzero number, or add a multiple of another row.
Three moves preserve the solution set: swap two rows; multiply a row by a nonzero scalar; or replace a row with itself plus a multiple of another row. Every elimination algorithm is made only from these moves.
Multiplying by zero is forbidden because it destroys a clue. Scaling only part of a row is forbidden because an equation includes its constant as well as its coefficients.
R₂ ← R₂ - 3R₁ is one operation. It means update every entry of row two, including the value to the right of the equals sign.
Challenge
Three Legal Moves
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Match the target matrix with the row operation that produces it.
Swap, nonzero scale, or add a multiple of another row. Apply the move across the entire row.
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The constant travels with its row
An augmented matrix keeps the whole equation together during row operations.
The system 2x + y = 5, x - y = 1 becomes [[2, 1 | 5], [1, -1 | 1]]. The bar is only a reading aid; every row operation crosses it.
The coefficient columns belong to variables. The final column belongs to the constants. Keeping them together prevents the classic error of simplifying the left side while leaving the right side unchanged.
A row [0, 0 | 3] says 0 = 3. The augmented column is what exposes a contradiction.
Challenge
Across the Bar
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Carry the same row operation through the constant column.
The bar separates roles, not arithmetic. Every move updates the whole row.
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Pivots build the staircase
Choose a nonzero leading entry, clear below it, then move down and right.
A pivot is the leading nonzero entry used to clear its column. If the next candidate is zero, swap in a row with a nonzero entry. Then eliminate below and repeat on the smaller lower-right block.
In row echelon form, every nonzero row starts farther right than the row above it, and all entries below each pivot are zero. Zero rows sit at the bottom. The matrix looks like a staircase.
A large pivot is not automatically better in exact arithmetic. In floating-point numerical code, libraries use pivoting strategies to reduce rounding error; the structural rule remains the same.
Challenge
Choose the Pivot
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Find the row that can supply the next pivot.
A pivot must be nonzero in the active column. These boards make the choice unambiguous.
Challenge
Read the Staircase
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Give the most specific description: REF, RREF, or neither.
RREF is also echelon, so choose RREF when its stronger rules hold.
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The staircase solves from the bottom up
The last pivot equation has the fewest unknowns, so solve it first.
Once the system is triangular, the bottom nonzero row isolates the last pivot variable. Solve it, substitute that value into the row above, and continue upward. That is back-substitution.
Reduced row echelon form goes farther: every pivot is scaled to one and its column is cleared above as well as below. RREF reads the solution directly, but ordinary echelon form plus back-substitution is already enough.
Do not divide early unless it helps. Keeping integer rows during forward elimination often makes the arithmetic easier, then the final pivots can be scaled at the end.
Challenge
Work Upward
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Solve a triangular system from its last pivot upward.
Start at the bottom row, then substitute each known value into the row above.
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Rank counts pivots, not rows
The staircase reveals independent information and all three solution outcomes.
The rank of a matrix is the number of pivot positions in its echelon form. A zero row contributes no new constraint, no matter how many rows the original matrix had.
For a system, compare coefficient pivots with the number of unknowns and inspect the augmented column. A pivot in the augmented column means no solution. Otherwise, fewer coefficient pivots than unknowns leaves free variables and infinitely many solutions. A pivot for every unknown gives one solution.
Gaussian elimination is simply the repeatable procedure that produces this staircase. Rank and the system verdict fall out of the same work used to solve it.
Challenge
Rank and Verdict
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Count coefficient pivots and classify the system.
A contradiction changes augmented rank. A missing coefficient pivot leaves a free variable.
Challenge
Gaussian Elimination
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Walk one system into staircase form, one exact move at a time.
Each board starts where the previous correct move ended. Pick the operation that clears the next entry.
