DPST1013 Chap.4 Linear Systems and Gaussian Elimination
Linear Systems and Gaussian Elimination
Gaussian elimination turns a collection of simultaneous linear equations into an equivalent system whose structure is easier to read. The method combines careful arithmetic with decisions about consistency, leading variables and free variables.
A reduced matrix is useful only when its rows are interpreted correctly: an all-zero coefficient row may express either a harmless identity or an impossible contradiction, depending on the augmented entry. This chapter develops a systematic route from the original equations to the complete solution set.
It covers numerical systems, parameter-dependent pivots, unknown right-hand sides and the connection between homogeneous solutions and translated families. The aim is to explain every possible solution, including cases where there are infinitely many or none.
A strong solution records valid row operations, keeps the augmented column attached, states parameter conditions before dividing, and substitutes the final result into the original equations. These habits make the working auditable and prevent a correct-looking partial reduction from hiding an incomplete case analysis.
What this chapter covers
- 01
Augmented matrices and reversible row operations
- 02
Echelon form, leading entries and back-substitution
- 03
Consistency and the number of free variables
- 04
Parameter-dependent pivots and separate cases
- 05
Particular solutions and homogeneous directions
- 06
Compatibility conditions for arbitrary right-hand sides
Let a symbolic pivot determine the solution set
- 2Subtract twice the first equation from the second, obtaining y+z=2. Subtract three times the first equation from the third, obtaining y+(1+k)z=2. These row replacements preserve the original solution set for every value of k because no division by a parameter has occurred.
- 2Subtract the new second equation from the new third to obtain kz=0. If k≠0, divide by k to find z=0. Then y=2 from y+z=2, and x=1 from the original first equation. This branch therefore has exactly one solution.
- 2If k=0, the last equation becomes an identity. Put z=t, giving y=2−t and x=1+t. The family is (1,2,0)+t(1,−1,1), with t real. Substitution gives 5, 12 and 17 on the three original left sides, confirming the whole family.
Key terms
- Augmented matrix
- A coefficient matrix with the right-hand vector appended as a final column. Each row represents one complete equation, so row operations must transform the augmented entry together with all its coefficients.
- Leading entry
- The first nonzero entry of a nonzero row in echelon form. Its column identifies a leading variable when it lies within the coefficient portion, or an inconsistency when it lies in the augmented column.
- Free variable
- A variable without a leading position in a consistent reduced system. It may be assigned a parameter, after which the leading variables are expressed in terms of that parameter through back-substitution.
- Consistent system
- A system with at least one solution. Consistency must be checked before counting free variables; a contradiction in one row eliminates all candidates regardless of the apparent freedom in other rows.
- Homogeneous system
- A system with zero on every right-hand side. It always has the zero solution, and any nonzero solutions describe directions that can be added to a particular solution of the corresponding nonhomogeneous system.
- Particular solution
- One specific vector satisfying the original nonhomogeneous equations. Adding all homogeneous solutions to this vector produces every solution, so it locates an affine family without describing all its freedom by itself.
- Compatibility condition
- A requirement on the right-hand side that makes a system solvable. It emerges when elimination produces a zero coefficient row whose transformed right-hand expression must also equal zero.
Linear Systems and Gaussian Elimination FAQ
Does a row of zeros prove that there are infinitely many solutions?
Not by itself. Inspect the augmented entry for consistency, then count the leading variable columns. A zero coefficient row with a nonzero right-hand entry is impossible. A harmless zero row only indicates redundancy; other rows may still determine every variable uniquely.
Why are row swaps allowed when solving a system?
Swapping rows only changes the order in which the same equations are displayed. It does not alter their simultaneous solutions. Row swaps are particularly useful when a planned pivot is zero but another row has a usable nonzero entry in that column.
Must every system be reduced all the way to reduced echelon form?
No. Ordinary echelon form followed by back-substitution is sufficient for solving a consistent system. Reduced echelon form may make free-variable expressions easier to read, but extra reduction is worthwhile only when it improves clarity or serves the question being asked.
How do I know whether my parameterised answer contains every solution?
Assign a parameter to each free variable and solve every leading equation in terms of those parameters. Explain that each solution must arise from some choice of the free variables, and substitute the symbolic family into the original equations to verify that every choice is allowed.
What happens if I divide by a symbolic pivot too early?
You implicitly exclude every parameter value making that pivot zero. Those excluded values may produce no solutions or infinitely many solutions. Separate the zero and nonzero cases before division, and interpret the entire final row in each branch.
Exam move
Practise reductions with an explicit row-operation ledger. After each operation, verify one eliminated coefficient and the updated augmented value before proceeding. This targets the two places where small arithmetic mistakes most often change a solution count. Next, take one consistent numerical system and change only its right-hand side. Predict which row will reveal inconsistency, then reduce it.
This exercise separates facts about the coefficient matrix from facts about a particular augmented system. For parameter questions, write the case split before calculating any reciprocal involving the parameter. For infinite families, label the particular vector and each homogeneous direction, then verify them separately.
Finally, rehearse a short explanation of why the number of free variables matches the number of independent parameters in your answer. A complete family is more than one example solution with an arbitrary symbol inserted.
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