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A $3\times3$ system row reduces to a unique solution

Example

Over $\mathbb R$ , the system $x+y+z=6,\qquad 2x-y+z=3,\qquad x+2y-z=2$ has the unique solution $(x,y,z)=(1,2,3)$ .

Facts & Assumptions

Given: The displayed real linear system.

[L1]

An RREF with a pivot in every variable column gives one solution and no free variables (Reduced row echelon form detects consistency and parametrises every solution by the nonpivot variables).

[L2]

Row operations on the augmented matrix preserve the solution set (Elementary row operations on an augmented matrix preserve exactly the solutions of its linear system).

Verification

technique · direct

1.1

Its augmented matrix reduces as follows: subtract twice row $0$ from row $1$ , subtract row $0$ from row $2$ , swap rows $1,2$ , add three times row $1$ to row $2$ , scale row $2$ by $-1/7$ , clear above its pivot, and then clear above the pivot in column $1$ ; the result is $[I_3\mid(1,2,3)^{\mathsf T}]$ .

L1L2algebra

2.1

The RREF gives the unique solution. Direct substitution yields $1+2+3=6$ , $2-2+3=3$ , and $1+4-3=2$ .

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §§1.2–1.3 (standard reference, not scraped)

A 3×33\times33×3 system row reduces to a unique solution