Solution of system of linear equations

At a Glance

• Frequency: 7 sub-parts across 7 of 13 years (2014, 2016, 2017, 2018, 2019, 2020, 2022)
• Priority tier: T2
• Marks (count): 7 (1), 10 (2), 13 (1), 15 (3)
• Average solve time: ~10 min
• Difficulty mix: medium 5, easy 2
• Section: A | Dominant type: computation

Why This Chapter Matters

Linear systems appear in every year from 2014 to 2022 (7 appearances) with consistent 10–15 marks in Section A. Five distinct archetypes rotate: direct row reduction, parameter-dependent consistency, solution via a given inverse, classifying a fixed system, and deriving a consistency condition. All reduce to one core operation — row-reduction of the augmented matrix — but the question varies from “solve it” to “classify it” to “find the parameter values”. The Rouché–Capelli theorem (rank of $A$ vs. rank of $[A|\mathbf b]$ vs. number of unknowns) is the organizing principle for three of the five archetypes.

Minimum Theory

Augmented matrix and row reduction. Represent $A\mathbf x=\mathbf b$ as $[A|\mathbf b]$ and apply elementary row operations to reach row-echelon (or reduced row-echelon) form.

Rouché–Capelli theorem. Let $r=\operatorname{rank}(A)$ and $\hat r=\operatorname{rank}([A|\mathbf b])$.

• $\hat r>r$: no solution (inconsistent — the last nonzero row gives $0=k\ne0$).
• $\hat r=r=n$ (number of unknowns): unique solution.
• $\hat r=r<n$: infinitely many solutions with $n-r$ free parameters.

Parametric system. If the system contains parameters $\lambda,\mu$, the reduced form will have a row whose left side depends on $\lambda$ and right side on $\mu$. Set the left side to zero to find critical $\lambda$; then check the right side to distinguish no solution from infinite solutions.

Solution via inverse. If $AB=cI$ is given, then $B^{-1}=A/c$ and the system $B\mathbf x=\mathbf b$ solves to $\mathbf x=(A/c)\mathbf b$. Match the coefficient matrix of the given system to $B$ (not $A$).

Question Archetypes

Five patterns cover every linear-systems question in the corpus.

solve-by-reduction (1 question(s); 2022)

Recognition Cues

• “Find all solutions to the system by [Gaussian/row-reduced] method.”
• No free parameters; the matrix has numerical entries only.

Solution Template

1. Write the augmented matrix $[A|\mathbf b]$.
2. Row-reduce to echelon form; note the rank.
3. If rank $=n$: back-substitute to get a unique solution.
4. If rank $<n$: set the $n-r$ free variables as parameters $t_1,t_2,\ldots$ and express the other variables in terms of them.

Worked Example(s)

2022 Paper 1, 2022-P1-Q2a (15 marks)

Solve $x_1+2x_2-x_3=2$, $2x_1+3x_2+5x_3=5$, $-x_1-3x_2+8x_3=-1$.

Row-reduce $[A|\mathbf b]$: $\begin{pmatrix}1&2&-1&|&2\\2&3&5&|&5\\-1&-3&8&|&-1\end{pmatrix}\to\begin{pmatrix}1&0&13&|&4\\0&1&-7&|&-1\\0&0&0&|&0\end{pmatrix}.$

Rank = 2, three unknowns → one free parameter. Let $x_3=t$: $\boxed{(x_1,x_2,x_3)=(4-13t,\;-1+7t,\;t),\quad t\in\mathbb R.}$

Common Traps

• Track the sign when negating a row ($R_2\to-R_2$ to get a leading $+1$).
• A row of zeros on both sides means one redundant equation — not no solution.

parametric-consistency (2 question(s); 2014, 2017)

Recognition Cues

• System with parameters $(\lambda,\mu)$; “investigate the values for which the system has (i) no solution (ii) unique (iii) infinite solutions.”
• Or: “for which $a$ does the system have a unique solution? For which $(a,b)$ does it have more than one?”

Solution Template

1. Row-reduce to echelon form; the last row will look like $(0\ 0\ \ldots\ |\,\lambda-c\ |\ \mu-d)$ or similar.
2. Unique: left-side pivot nonzero (so rank = $n$). State the condition: $\lambda\ne c$.
3. No solution: left side zero but right side nonzero: $\lambda=c$ and $\mu\ne d$.
4. Infinite: both sides zero: $\lambda=c$ and $\mu=d$.

Worked Example(s)

2014 Paper 1, 2014-P1-Q2bi (10 marks)

$x+y+z=6$, $x+2y+3z=10$, $x+2y+\lambda z=\mu$. Classify by $(\lambda,\mu)$.

Row-reduce: last row becomes $(0\ 0\ |\,\lambda-3\ |\ \mu-10)$.

• Unique: $\lambda\ne3$ (any $\mu$).
• No solution: $\lambda=3$, $\mu\ne10$.
• Infinite: $\lambda=3$, $\mu=10$.

2017 Paper 1, 2017-P1-Q4b (15 marks)

$x+2y+2z=1$, $x+ay+3z=3$, $x+11y+az=b$. Find $a$ for unique; find $(a,b)$ for infinite.

$\det A=(a-5)(a+1)$. Unique: $a\ne5$ and $a\ne-1$. Infinite: $(a,b)=(-1,-5)$ or $(5,7)$.

Common Traps

• Test consistency (last row of augmented) separately from uniqueness (last row of coefficient). "$\det A=0$" only rules out uniqueness; you must still check whether the augmented system is consistent.
• State all three cases explicitly even if asked for only one or two.

classify-solutions (1 question(s); 2018)

Recognition Cues

• “Determine which of the following is true: (i) no solution (ii) unique (iii) infinite.”
• The system is fully numerical; classification only, with explanation.

Solution Template

1. Compute $\det A$. If $\det A\ne0$: unique solution; done.
2. If $\det A=0$: row-reduce $[A|\mathbf b]$. Compare $\operatorname{rank}(A)$ and $\operatorname{rank}([A|\mathbf b])$.
3. Equal ranks $<n$: infinite. Unequal: no solution.

Worked Example(s)

2018 Paper 1, 2018-P1-Q3a (13 marks)

$x+3y-2z=-1$, $5y+3z=-8$, $x-2y-5z=7$. Classify.

$\det A=-19+19=0$. Row-reduce $[A|\mathbf b]$: last row $\to(0\ 0\ 0\ |\ 0)$. Rank$(A)=$ Rank$([A|\mathbf b])=2<3$.

(i) False. (ii) False. (iii) True — infinitely many solutions.

Common Traps

• $\det A=0$ means no unique solution; it does not mean no solution at all. Must test augmented matrix.

solve-via-inverse (2 question(s); 2019, 2020)

Recognition Cues

• “Show that $AB=cI$; use this result to solve the system […].”
• The coefficient matrix of the given system matches one of $A$ or $B$ (usually $B$).

Solution Template

1. Verify $AB=cI$ by direct matrix multiplication (show each entry).
2. Conclude $B^{-1}=A/c$ (or $A^{-1}=B/c$).
3. Match the given system to $B\mathbf x=\mathbf b$ (align rows of $B$ with the system’s equations).
4. Solve: $\mathbf x=B^{-1}\mathbf b=(A/c)\mathbf b$.

Worked Example(s)

2019 Paper 1, 2019-P1-Q1d (10 marks)

$A=\bigl[\begin{smallmatrix}1&2&1\\1&-4&1\\3&0&-3\end{smallmatrix}\bigr]$, $B=\bigl[\begin{smallmatrix}2&1&1\\1&-1&0\\2&1&-1\end{smallmatrix}\bigr]$. Show $AB=6I_3$. Solve $2x+y+z=5$, $x-y=0$, $2x+y-z=1$.

Verify $AB=6I_3$ by row-by-row multiplication. The system’s coefficient matrix is $B$; $\mathbf b=(5,0,1)^T$.

$\mathbf x=\tfrac16 A\mathbf b=\tfrac16(6,6,12)^T=(1,1,2)^T$.

$\boxed{x=1,\quad y=1,\quad z=2.}$

Common Traps

• Match the system to $B$, not $A$. If you set up $A\mathbf x=\mathbf b$ instead, you get $\mathbf x=(B/c)\mathbf b$, which is wrong.
• Confirm $AB=cI$ means $B^{-1}=A/c$ AND $A^{-1}=B/c$ (they are two-sided inverses in finite dimensions).

consistency-condition (1 question(s); 2016)

Recognition Cues

• “Using elementary row operations, find the condition that the system has a solution.”
• The right-hand side contains a free parameter; find the constraint that prevents the last row from giving $0=k\ne0$.

Solution Template

1. Row-reduce $[A|\mathbf b]$ symbolically.
2. Read the last row’s right-side entry after reduction; set it to $0$.
3. State the condition.

Common Traps

• Keep the symbolic right-hand side throughout all row operations — students often lose the parameter mid-reduction.

Practice Set

• 2020-P1-Q4a (15 m) — — compute $AB$, find $\det$, use inverse to solve; combines verification with application.
• 2016-P1-Q1b (7 m) — — consistency condition via row reduction; short but conceptually important.