NDA Maths · Matrices & Determinants

Linear Systems: Consistency & Cramer's Rule

A square linear system AX = B is solved and classified by one number — the coefficient determinant: nonzero gives a unique Cramer's-rule solution, zero gives either no solution or infinitely many.

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Why this matters

Eight PYQs — small but a reliable scorer and the payoff of the whole chapter: determinants and inverses applied to solving equations. Questions test consistency from the coefficient determinant, Cramer's rule, homogeneous systems, and finding the parameter k that makes a system consistent or not. Four concepts cover it.

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Concept 1 of 4

Consistency from the coefficient determinant

Intuition

For a square system

AX = B

, the coefficient determinant

|A|

tells you everything about the solution count BEFORE you solve:

|A| \neq 0

means exactly one solution;

|A| = 0

means either no solution or infinitely many.

Definition

For $AX = B$ (square): if $|A| \neq 0$ → unique solution. If $|A| = 0$ : the system is inconsistent (no solution) or has infinitely many solutions, decided by whether the equations are genuinely contradictory or just dependent (e.g. two equations the same scaling but different constants → no solution).

Worked example

Classify the system

x+y+z=6,\ x-y+z=2,\ 2x+y-z=1

Form the coefficient determinant $|A| = \begin{vmatrix}1&1&1\\1&-1&1\\2&1&-1\end{vmatrix}$ .
Expand along row 1: $1(1-1) - 1(-1-2) + 1(1+2) = 0 + 3 + 3 = 6$ .
$|A| = 6 \neq 0$ , so the coefficient matrix is non-singular.

Answer:Unique solution — consistent and independent (

|A| = 6 \neq 0

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 1Matrices & DeterminantsEASY

The system of linear equations

x+2y+z=4

2x+4y+2z=8

and

3x+6y+3z=10

has

[Q8 · Apr · 2023]

Drill 2 more on consistency from the coefficient determinant

Concept 2 of 4

Cramer's rule

Intuition

When

|A| \neq 0

, each unknown is a ratio of determinants: replace the column of coefficients for that variable with the constants column, take the determinant, and divide by

|A|

Definition

For $AX = B$ with $|A| = \Delta \neq 0$ : $x = \Delta_x/\Delta$ , $y = \Delta_y/\Delta$ , $z = \Delta_z/\Delta$ , where $\Delta_x$ is $\Delta$ with the $x$ -column replaced by $B$ (and similarly for $\Delta_y, \Delta_z$ ).

Cramer's rule

x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}

Worked example

Solve

x + y = 3,\ x - y = 1

by Cramer's rule.

$\Delta = \begin{vmatrix}1&1\\1&-1\end{vmatrix} = -2$ .
$\Delta_x = \begin{vmatrix}3&1\\1&-1\end{vmatrix} = -4$ ; $\Delta_y = \begin{vmatrix}1&3\\1&1\end{vmatrix} = -2$ .
$x = \tfrac{-4}{-2} = 2,\ y = \tfrac{-2}{-2} = 1$ .

Answer:

x = 2,\ y = 1

Practice this concept4 quick reps

From the bank · past-year question

Example 2Matrices & DeterminantsMODERATE

In obtaining the solution of the system of equations

x+y+z=7

x+2y+3z=16

and

x+3y+4z=22

by Cramer's rule, the value of

y

is obtained by dividing

D

D_2

, where

D=\begin{vmatrix}1&1&1\\1&2&3\\1&3&4\end{vmatrix}

. What is the value of the determinant

D_2

[Q14 · Sep · 2025]

Drill 1 more on cramer's rule

Concept 3 of 4

Homogeneous systems and the solution space

Intuition

A homogeneous system

AX = 0

always has the trivial solution

X = 0

. It has extra (non-trivial) solutions exactly when

|A| = 0

. And if a non-homogeneous system has two distinct solutions, it must have infinitely many.

Definition

$AX = 0$ has a non-trivial solution iff $|A| = 0$ (otherwise only $X = 0$ ). For $AX = B$ : if $X_1 \neq X_2$ are both solutions, then $X_1 + t(X_2 - X_1)$ is a solution for every $t$ → infinitely many (the solution set is never exactly two).

Worked example

AX = B

has two distinct solutions

X_1

and

X_2

, how many solutions does it have?

$X_1$ and $X_2$ solve it, so $A(X_1 - X_2) = 0$ — a non-trivial solution of the homogeneous system.
Then $X_1 + t(X_2 - X_1)$ solves $AX = B$ for every scalar $t$ .
That's an infinite family.

Answer:Infinitely many solutions.

Practice this concept4 quick reps

From the bank · past-year question

Example 3Matrices & DeterminantsMODERATE

Let

AX=B

be a system of 3 linear equations with 3-unknowns. Let

X_1

and

X_2

be its two distinct solutions. If the combination

aX_1+bX_2

is a solution of

AX=B

; where

a

b

are real numbers, then which one of the following is correct?

[Q9 · Apr · 2023]

Concept 4 of 4

Finding the parameter for consistency

Intuition

When a coefficient (or constant) contains a parameter

k

, the borderline cases live where

|A| = 0

. Set the coefficient determinant to 0 to find the candidate

k

values, then check each to see if it gives no solution or infinitely many.

Definition

Put the parameter into $|A|$ and solve $|A| = 0$ for the critical $k$ . For each such $k$ , substitute back: dependent-and-consistent → infinitely many; contradictory → no solution. Away from those $k$ , the solution is unique.

Worked example

For what

k

does

kx + y + z = 1,\ x + ky + z = k,\ x + y + kz = k^2

have no solution?

Coefficient determinant $= (k-1)^2(k+2)$ (standard symmetric system).
It is 0 at $k = 1$ and $k = -2$ .
At $k = 1$ all three equations coincide → infinitely many. At $k = -2$ they're contradictory → no solution.

Answer:

k = -2

(at

k = 1

there are infinitely many instead).

Practice this concept4 quick reps

From the bank · past-year question

Example 4Matrices & DeterminantsHARD

The system of equations

kx + y + z = 1

x + ky + z = k

and

x + y + kz = k^2

has no solution if

k

equals

[Q27 · Sep · 2017]

Drill 1 more on finding the parameter for consistency

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Cramer's rule
Cramer's rule
$x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}$

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