Linear Equations And Cramer’s Rule

MathsDeterminantsFor JEE aspirants

Cramer's Rule is a formula that expresses the unique solution of a non-homogeneous system of linear equations in terms of determinants: $x = Δ_{x} /Δ$ , $y = Δ_{y} /Δ$ , $z = Δ_{z} /Δ$ , valid whenever $Δ \neq = 0$ . For homogeneous systems (right-hand side zero), the system has a non-trivial solution if and only if $Δ = 0$ . This concept, essential for JEE Mathematics, ties together determinants and linear equations by classifying every $3 \times 3$ system as consistent-independent, consistent-dependent, or inconsistent based on the values of $Δ$ and the auxiliary determinants $Δ_{x}$ , $Δ_{y}$ , $Δ_{z}$ .

Key Formulas - Quick Reference

Homogeneous system $a_{i} x + b_{i} y + c_{i} z = 0$ ( $i = 1, 2, 3$ ): non-trivial solution exists iff $Δ = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} = 0$
If $Δ \neq = 0$ : only the trivial solution $x = y = z = 0$ exists.
Non-homogeneous system $a_{i} x + b_{i} y + c_{i} z = d_{i}$ with $Δ \neq = 0$ : $x = \frac{Δ _{x}}{Δ}, y = \frac{Δ _{y}}{Δ}, z = \frac{Δ _{z}}{Δ}$
Auxiliary determinants: $Δ_{x}$ replaces column of $x$ -coefficients with the constants column $(d_{1}, d_{2}, d_{3})$ ; similarly for $Δ_{y}$ and $Δ_{z}$ .
Consistency test: unique solution when $Δ \neq = 0$ ; infinitely many solutions when $Δ = Δ_{x} = Δ_{y} = Δ_{z} = 0$ ; no solution when $Δ = 0$ but at least one of $Δ_{x}, Δ_{y}, Δ_{z} \neq = 0$ .

1. Homogeneous Linear Equations

Consider the system of homogeneous simultaneous linear equations in three unknowns:

$a_{1} x + b_{1} y + c_{1} z = 0$

$a_{2} x + b_{2} y + c_{2} z = 0$

$a_{3} x + b_{3} y + c_{3} z = 0$

Here every constant term is zero. The obvious solution is $x = y = z = 0$ , called the trivial solution.

The homogeneous system has a non-trivial solution (i.e. at least one of

x, y, z

is nonzero) if and only if

Δ = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} = 0.

Case Summary for the Homogeneous System

Condition	Solution	Terminology
$Δ \neq = 0$	$x = y = z = 0$ only	Trivial solution
$Δ = 0$	Infinitely many solutions (including the trivial one)	Non-trivial solution exists

Solved Example 1

Problem: If the system of equations

3 x + 10 y + 17 z = 0

- x + 6 y + 13 z = 0

, and

20 x - 13 y + λ z = 0

has a non-trivial solution, find the value of

λ

and the solution.

Solution:

A non-trivial solution requires $Δ = 0$ . Inspecting the coefficient patterns:

Row 1: $3, 10, 17$ - common difference $7$ (A.P.)
Row 2: $- 1, 6, 13$ - common difference $7$ (A.P.)
Row 3: $20, - 13, λ$ - for A.P., common difference is $- 33$ , so $λ = - 13 + (- 33) = - 46$ .

By Property 9 (A.P. property), with $λ = - 46$ all three rows are in A.P., forcing $Δ = 0$ .

To find the solution, let $z = k$ (a free parameter). Substituting into the first two equations:

$3 x + 10 y = - 17 k - x + 6 y = - 13 k$

Multiply the second by $3$ : $- 3 x + 18 y = - 39 k$ . Add to the first: $28 y = - 56 k$ , so $y = - 2 k$ . Substituting back: $3 x + 10 (- 2 k) = - 17 k \Rightarrow 3 x = 3 k \Rightarrow x = k$ .

Therefore $(x, y, z) = (k, - 2 k, k)$ for any $k \neq = 0$ . So $λ = - 46$ and the solution family is $(k, - 2 k, k)$ .

2. Non-Homogeneous Systems: Cramer's Rule

Consider a non-homogeneous system with $Δ \neq = 0$ :

$a_{1} x + b_{1} y + c_{1} z = d_{1}$

$a_{2} x + b_{2} y + c_{2} z = d_{2}$

$a_{3} x + b_{3} y + c_{3} z = d_{3}$

Cramer's Rule: If

Δ = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} \neq = 0

and

(d_{1}, d_{2}, d_{3}) \neq = (0, 0, 0)

, then the unique solution is

x = \frac{Δ _{x}}{Δ}, y = \frac{Δ _{y}}{Δ}, z = \frac{Δ _{z}}{Δ}

where

Δ_{x} = d_{1} d_{2} d_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}, Δ_{y} = a_{1} a_{2} a_{3} d_{1} d_{2} d_{3} c_{1} c_{2} c_{3}, Δ_{z} = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} d_{1} d_{2} d_{3}

Each

Δ_{i}

is obtained from

Δ

by replacing the corresponding coefficient column with the constants column.

3. Consistency of Linear Systems

A non-homogeneous system can have exactly one solution, infinitely many solutions, or no solution. The classification depends on $Δ$ and the auxiliary determinants.

Condition	Solution Type	Terminology
$Δ \neq = 0$ (any $Δ_{x}, Δ_{y}, Δ_{z} \in R$ )	Unique solution given by Cramer's rule	Consistent, independent
$Δ = 0$ and $Δ_{x} = Δ_{y} = Δ_{z} = 0$	Infinitely many solutions	Consistent, dependent
$Δ = 0$ and at least one of $Δ_{x}, Δ_{y}, Δ_{z} \neq = 0$	No solution	Inconsistent

Note: The condition "

Δ_{x} = Δ_{y} = Δ_{z} = 0

" together with

Δ = 0

is a necessary condition for infinite solutions but must be checked against the original equations for consistency, since a rank mismatch can still make the system inconsistent in rare degenerate cases.

Solved Example 2

Problem: Prove that the system of equations

237 x + 229 y + 221 z = 213

- 3 x + 7 y + 17 z = 27

, and

7 x + 19 y + 31 z = 43

has infinitely many solutions.

Solution:

Examine the coefficients and constants row by row:

Row 1: $237, 229, 221, 213$ - common difference $- 8$
Row 2: $- 3, 7, 17, 27$ - common difference $10$
Row 3: $7, 19, 31, 43$ - common difference $12$

Each row (including the constant term) is in A.P. with common differences $- 8$ , $10$ , $12$ respectively.

For the coefficient determinant $Δ$ : rows are $(237, 229, 221)$ , $(- 3, 7, 17)$ , $(7, 19, 31)$ . Each row is an A.P., so by Property 9, $Δ = 0$ .

Similarly for $Δ_{x}$ (replace first column with constants): rows are $(213, 229, 221)$ , $(27, 7, 17)$ , $(43, 19, 31)$ . Each row is still in A.P. (constants are chosen so this holds), so $Δ_{x} = 0$ . The same reasoning gives $Δ_{y} = 0$ and $Δ_{z} = 0$ .

Since $Δ = Δ_{x} = Δ_{y} = Δ_{z} = 0$ , the system is consistent-dependent with infinitely many solutions.

Solved Example 3

Problem: For what values of

p

and

q

does the system

2 x + p y + 6 z = 8, x + 2 y + q z = 5, x + y + 3 z = 4

have (i) no solution, (ii) a unique solution, (iii) infinitely many solutions?

Solution:

Compute $Δ$ :

$Δ = 211 p 21 6 q 3 = 2 (6 - q) - p (3 - q) + 6 (1 - 2)$

$= 12 - 2 q - 3 p + pq - 6 = pq - 2 q - 3 p + 6 = (p - 2) (q - 3)$

Compute $Δ_{1}$ (replace column 1 with constants $8, 5, 4$ ):

$Δ_{1} = 854 p 21 6 q 3 = 8 (6 - q) - p (15 - 4 q) + 6 (5 - 8)$

$= 48 - 8 q - 15 p + 4 pq - 18 = 4 pq - 8 q - 15 p + 30 = (4 q - 15) (p - 2)$

Compute $Δ_{2}$ (replace column 2):

$Δ_{2} = 211854 6 q 3 = 2 (15 - 4 q) - 8 (3 - q) + 6 (4 - 5) = 30 - 8 q - 24 + 8 q - 6 = 0$

Compute $Δ_{3}$ (replace column 3):

$Δ_{3} = 211 p 21 854 = 2 (8 - 5) - p (4 - 5) + 8 (1 - 2) = 6 + p - 8 = p - 2$

Case I: When $q = 3$ and $p \neq = 2$ : $Δ = 0$ , $Δ_{1} = (4 (3) - 15) (p - 2) = - 3 (p - 2) \neq = 0$ . So the system has no solution.

Case II: When $Δ \neq = 0$ , i.e. $p \neq = 2$ and $q \neq = 3$ : the system has a unique solution (given by Cramer's rule).

Case III: When $p = 2$ : $Δ = 0$ , and $Δ_{1} = (4 q - 15) (0) = 0$ , $Δ_{2} = 0$ , $Δ_{3} = 0$ . So the system has infinitely many solutions.

(When $q = 3$ and $p = 2$ , both $Δ = 0$ and all $Δ_{i} = 0$ , so this also gives infinitely many solutions - a sub-case of Case III.)

Common Mistakes to Avoid

Watch out

Confusing homogeneous and non-homogeneous cases. For a homogeneous system, $Δ = 0$ gives infinitely many solutions (including trivial). For a non-homogeneous system, $Δ = 0$ requires you to check $Δ_{x}, Δ_{y}, Δ_{z}$ before concluding.
Using Cramer's rule when $Δ = 0$ . Cramer's rule gives $x = Δ_{x} /Δ$ , which is undefined when $Δ = 0$ . Switch to consistency analysis instead.
Forgetting to replace the correct column. $Δ_{x}$ replaces the x-coefficient column (i.e. column 1), $Δ_{y}$ replaces column 2, and $Δ_{z}$ replaces column 3. Mixing these up will give the wrong sign or wrong value.
Concluding "no solution" from $Δ = 0$ alone. $Δ = 0$ can mean either no solution or infinitely many solutions - the distinction is decided by whether the auxiliary determinants are also zero.
Applying Cramer's rule to systems with more unknowns than equations, or vice versa. Cramer's rule applies only to square systems ( $n$ equations in $n$ unknowns). For non-square systems, use the augmented-matrix rank method (covered in the Matrices concept).
Ignoring the A.P. property as a fast diagnostic. If the coefficient rows are all in A.P., $Δ = 0$ automatically - saving you from a full expansion.

Frequently Asked Questions

When does a homogeneous system have a non-trivial solution?

A homogeneous system $A X = 0$ has a non-trivial solution (i.e. at least one variable nonzero) if and only if the coefficient determinant $Δ = 0$ . If $Δ \neq = 0$ , the only solution is the trivial one $x = y = z = 0$ .

Can Cramer's rule be used when the determinant is zero?

No. Cramer's rule gives $x = Δ_{x} /Δ$ , which is undefined when $Δ = 0$ . In that case, you must check whether all of $Δ_{x}, Δ_{y}, Δ_{z}$ are also zero: if yes, the system has infinitely many solutions; if no, it has no solution.

What is the difference between consistent-dependent and consistent-independent?

Consistent-independent means the system has a unique solution (this happens when $Δ \neq = 0$ ). Consistent-dependent means the system has infinitely many solutions - the equations are not independent, so at least one is a linear combination of the others (this happens when $Δ = Δ_{x} = Δ_{y} = Δ_{z} = 0$ ).

How do I recognise an inconsistent system quickly?

An inconsistent system in three variables satisfies $Δ = 0$ but at least one of $Δ_{x}, Δ_{y}, Δ_{z} \neq = 0$ . Geometrically, this corresponds to three planes with no common point of intersection - either two are parallel or all three form a triangular prism configuration.

Why does Cramer's rule work?

Cramer's rule follows from the row-column properties of determinants. If $A X = D$ , multiplying both sides by $A^{- 1}$ gives $X = A^{- 1} D$ . The entries of $A^{- 1} D$ can be written using the adjoint of $A$ and expanded to give the Cramer's formulas. It is essentially the determinantal form of the matrix inverse method.

Is Cramer's rule efficient for large systems?

For $n \geq 4$ , Cramer's rule becomes computationally expensive because each $Δ_{i}$ requires evaluating an $n \times n$ determinant. Gaussian elimination and matrix inversion are far more efficient. However, for $2 \times 2$ and $3 \times 3$ JEE problems, Cramer's rule is often the fastest method by hand.

What is the geometric meaning of $Δ = 0$ in a linear system?

$Δ = 0$ means the coefficient vectors are linearly dependent, so the three planes represented by the equations are either coincident (infinitely many common points), parallel (no common points), or intersecting in a line (infinitely many points). The auxiliary determinants distinguish these cases.

How does the A.P. property help solve linear-equation problems?

If the coefficient rows (or columns) of a system are in arithmetic progression, $Δ = 0$ automatically by Property 9 of determinants - no expansion needed. This is a huge shortcut in JEE Advanced problems where the coefficients follow an obvious pattern.

Previous year questions on Linear Equations And Cramer’s Rule

22 questions from past papers, each with a step-by-step solution.

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1. Homogeneous Linear Equations

Case Summary for the Homogeneous System

2. Non-Homogeneous Systems: Cramer's Rule

3. Consistency of Linear Systems

Common Mistakes to Avoid

Frequently Asked Questions

When does a homogeneous system have a non-trivial solution?

Can Cramer's rule be used when the determinant is zero?

What is the difference between consistent-dependent and consistent-independent?

How do I recognise an inconsistent system quickly?

Why does Cramer's rule work?

Is Cramer's rule efficient for large systems?

What is the geometric meaning of Δ=0 in a linear system?

How does the A.P. property help solve linear-equation problems?

Previous year questions on Linear Equations And Cramer’s Rule

Ready to master Determinants?

What is the geometric meaning of $Δ = 0$ in a linear system?