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JEE Mains Maths · Formula sheet

Determinants formulas

16 formulas and 16 common traps for JEE Mains Maths Determinants, grouped by subtopic.

Full notes with worked examples

Infinitely Many Solutions: One Equation Holds the Parameters

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The third equation as a combination

Dependent third equation

E3=pE1+qE2E_3=pE_1+qE_2

When the determinant vanishes for every value

Dependent rows

R3=pR1+qR2 ⇒ consistent exactly when b3=pb1+qb2R_3=pR_1+qR_2\ \Rightarrow\ \text{consistent exactly when } b_3=pb_1+qb_2

Common traps

Match the right-hand side too

Matching the coefficients alone gives Δ=0\Delta=0, which also allows no solution. The right-hand side of the third equation must be the same combination of the other two.

Look for dependence before expanding

Expanding a determinant with a parameter and finding it identically 0 wastes time. Check first whether a row is a sum or multiple of others.

Infinitely Many Solutions: Parameters in Two Equations

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The coefficient determinant and one Cramer determinant

Cramer's rule

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

Elimination instead of determinants

The last equation decides

kz=c:k≠0 unique; k=c=0 infinite; k=0≠c nonekz=c:\quad k\neq0\ \text{unique};\ k=c=0\ \text{infinite};\ k=0\neq c\ \text{none}

Common traps

Confirm consistency

Δ=0\Delta=0 with one vanishing Cramer determinant is the usual shortcut, but the definition asks for all of them to vanish. When two options differ only in the right-hand constant, confirm by eliminating once.

Keep the right-hand sides

Elimination must carry the constants along. Dropping them loses the difference between infinitely many solutions and none.

No Solution and Inconsistent Systems

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The value that gives no solution

Inconsistency

Δ=0,(Δx,Δy,Δz)≠(0,0,0)\Delta=0,\quad(\Delta_x,\Delta_y,\Delta_z)\neq(0,0,0)

Counting the values

Count what survives

#{θ:Δ(θ)=0, inconsistent}\#\{\theta:\Delta(\theta)=0,\ \text{inconsistent}\}

Common traps

A root of the determinant may give infinitely many

Each value with Δ=0\Delta=0 must be tested. In kx+y=1, x+ky=1kx+y=1,\ x+ky=1, k=1k=1 gives the same line twice, and only k=−1k=-1 gives no solution.

Candidates are not answers

Every root of Δ\Delta is only a candidate. Discard those that give infinitely many solutions before counting.

Classifying a System: Unique, Infinite or None

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The full classification

Three cases

Δ≠0⇒unique;Δ=0⇒infinite or none\Delta\neq0\Rightarrow\text{unique};\quad\Delta=0\Rightarrow\text{infinite or none}

The unique solution

Cramer's rule

x=ΔxΔ, y=ΔyΔ, z=ΔzΔ(Δ≠0)x=\frac{\Delta_x}{\Delta},\ y=\frac{\Delta_y}{\Delta},\ z=\frac{\Delta_z}{\Delta}\quad(\Delta\neq0)

Common traps

Two options can be wrong

In 'which is NOT correct' questions, check every option against the table, not just until one fails. A booklet may print two incorrect statements; the key then picks one.

Count ordered choices

With two dice, the pairs (a,b)(a,b) and (b,a)(b,a) are different outcomes. Count the pairs making Δ=0\Delta=0 as ordered pairs out of 36.

Homogeneous Systems and Non-trivial Solutions

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An angle in the coefficients

Non-trivial solutions

Ax=0 has x≠0 exactly when ∣A∣=0A\mathbf x=\mathbf0\ \text{has }\mathbf x\neq\mathbf0\ \text{exactly when}\ |A|=0

An algebraic constant

Symmetric determinant

∣k111k111k∣=(k+2)(k−1)2\begin{vmatrix}k&1&1\\1&k&1\\1&1&k\end{vmatrix}=(k+2)(k-1)^2

Common traps

Open or closed interval

Trig solutions often sit at the ends, 00 or π\pi. Check whether the interval includes them before counting.

Every root counts

(k+2)(k−1)2=0(k+2)(k-1)^2=0 has two distinct values. A question asking for the sum or number of values needs both.

Determinants of kA, adj A and Products

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Scalar multiples and adjoints

Adjoint rules

∣kA∣=kn∣A∣,∣adjA∣=∣A∣n−1|kA|=k^n|A|,\qquad|\mathrm{adj}A|=|A|^{n-1}

Products, inverses and cofactors

Product rule

∣AB∣=∣A∣ ∣B∣|AB|=|A|\,|B|

Common traps

The power depends on the order

∣kA∣=kn∣A∣|kA|=k^n|A|, not k∣A∣k|A|, and ∣adjA∣=∣A∣n−1|\mathrm{adj}A|=|A|^{n-1} changes with nn. Read the order of the matrix before applying either.

Divide by a negative determinant

When ∣A∣|A| is negative, formulas like ∣X∣=∣AX∣∣A∣|X|=\frac{|AX|}{|A|} flip the sign. Stopping at ∣A∣∣X∣|A||X| gives the wrong sign.

Simplifying Determinants by Row and Column Operations

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Factor out and create zeros

Row operation

Ri→Ri−kRj: Δ unchangedR_i\to R_i-kR_j:\ \Delta\ \text{unchanged}

Expand and compare

Vandermonde

∣1aa21bb21cc2∣=(a−b)(b−c)(c−a)\begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix}=(a-b)(b-c)(c-a)

Common traps

Scaling a row scales the determinant

Replacing RiR_i by 2Ri−Rj2R_i-R_j doubles the determinant. Only adding a multiple of another row is free.

Sign of the cofactor

Expanding along a row, the signs alternate +,−,++,-,+ starting from the top-left. A middle-column term carries a minus sign.

Determinants as Functions of x

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Trigonometric determinants

Range of a sinusoid

asin⁡θ+bcos⁡θ∈[−a2+b2, a2+b2]a\sin\theta+b\cos\theta\in\left[-\sqrt{a^2+b^2},\ \sqrt{a^2+b^2}\right]

Derivatives, integrals and limits

Derivative of a determinant

ddx∣f1f2g1g2∣=∣f1′f2′g1g2∣+∣f1f2g1′g2′∣\frac{d}{dx}\begin{vmatrix}f_1&f_2\\g_1&g_2\end{vmatrix}=\begin{vmatrix}f_1'&f_2'\\g_1&g_2\end{vmatrix}+\begin{vmatrix}f_1&f_2\\g_1'&g_2'\end{vmatrix}

Common traps

Simplify before expanding

Expanding a trigonometric determinant straight away produces long products that are easy to get wrong. One row operation or identity usually removes most terms first.

Differentiate one row at a time

The derivative of a determinant is not the determinant of the derivatives. Differentiate each row separately and add, or expand first.

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