JEE Maths — Matrices and Determinants Complete Chapter Guide

Chapter Overview & Weightage

Matrices and Determinants is one of the most reliable scoring topics in JEE Main. Unlike Calculus where a single wrong substitution can derail a 6-mark problem, this chapter rewards systematic students — the steps are predictable, and the marks are consistent.

JEE Main Weightage: 5–7% (2–3 questions per paper)

This chapter has appeared in every single JEE Main session for the past 5 years. Expect at minimum one determinant property question and one system-of-equations/inverse question per shift.

Year	JEE Main (Questions)	Marks	JEE Advanced
2024	2–3 per shift	8–12	1 (matrix equation)
2023	2 per shift	8	1 (system of equations)
2022	2–3 per shift	8–12	1–2
2021	2 per shift	8	1
2020	2 per shift	8	1

For JEE Advanced, the questions are trickier — often combining matrices with functions or abstract algebra. But JEE Main questions follow a very tight pattern, which we’ll use to our advantage.

Key Concepts You Must Know

Ranked by how often they appear in PYQs:

Tier 1 — Almost Certain to Appear

Properties of determinants (row/column operations, factor theorem)
Finding inverse using adjoint: $A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A)$
System of linear equations — consistency conditions
Characteristic equations and Cayley-Hamilton theorem

Tier 2 — High Probability

Matrix multiplication and its properties (non-commutativity)
Symmetric, skew-symmetric, orthogonal matrices
Eigenvalues (sum = trace, product = determinant)
Cramer’s rule for 2×2 and 3×3 systems

Tier 3 — Know the Basics

Elementary row/column operations
Rank of a matrix
Idempotent, involutory, nilpotent matrices

Important Formulas

|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})

When to use: Any time you need $|A|$ from scratch. Always expand along the row or column with the most zeros — saves enormous calculation time.

\text{adj}(A) = [C_{ij}]^T \quad \text{(transpose of cofactor matrix)}

A^{-1} = \frac{\text{adj}(A)}{|A|}, \quad |A| \neq 0

When to use: Whenever you need to solve $AX = B$ or find $A^{-1}$ explicitly. For 2×2:

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

|kA| = k^n |A| \quad \text{(for } n \times n \text{ matrix)}

|AB| = |A||B|

|A^T| = |A|

|\text{adj}(A)| = |A|^{n-1}

\text{adj}(\text{adj}(A)) = |A|^{n-2} \cdot A

When to use: JEE loves asking $|\text{adj}(A)|$ or $|\text{adj}(\text{adj}(A))|$ — memorize these two cold.

Every matrix satisfies its own characteristic equation.

For 2×2: if characteristic equation is $\lambda^2 - (\text{tr}A)\lambda + |A| = 0$ , then:

A^2 - (\text{tr}A)A + |A|I = 0

When to use: When asked to compute high powers of a matrix ( $A^{10}$ , $A^{100}$ ), or when given a matrix equation involving $A^2$ .

For $AX = B$ :

$|A| \neq 0$ → unique solution: $X = A^{-1}B$
$|A| = 0$ and $(\text{adj}A)B = 0$ → infinitely many solutions
$|A| = 0$ and $(\text{adj}A)B \neq 0$ → no solution (inconsistent)

When to use: The JEE Main 2024 Shift 2 question asked exactly this — given a parameter $k$ , find values for which the system has infinite solutions.

Solved Previous Year Questions

PYQ 1 — JEE Main 2024 (Shift 1)

Question: If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ , find $|\text{adj}(\text{adj}(A))|$ .

Solution:

First compute $|A|$ :

|A| = (1)(4) - (2)(3) = 4 - 6 = -2

Now use the formula $|\text{adj}(\text{adj}(A))| = |A|^{(n-1)^2}$ for an $n \times n$ matrix.

For $n = 2$ : $(n-1)^2 = 1$ , so:

|\text{adj}(\text{adj}(A))| = |A|^1 = -2

Wait — let’s be careful here. The cleaner approach uses the chain:

|\text{adj}(A)| = |A|^{n-1} = (-2)^{2-1} = -2

|\text{adj}(\text{adj}(A))| = |\text{adj}(A)|^{n-1} = (-2)^1 = -2

Answer: $-2$

Students confuse $|\text{adj}(A)| = |A|^{n-1}$ with $|\text{adj}(A)| = |A|^n$ . The exponent is $n-1$ , not $n$ . Write this on your formula sheet right now.

PYQ 2 — JEE Main 2023 (Shift 2)

Question: For what value of $k$ does the system $x + y + z = 1$ , $2x + 3y + 2z = k$ , $5x + 9y + kz = 4$ have no solution?

Solution:

Write the coefficient matrix and find when $|A| = 0$ :

A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 3 & 2 \\ 5 & 9 & k \end{pmatrix}

Expanding along Row 1:

|A| = 1(3k - 18) - 1(2k - 10) + 1(18 - 15)

= 3k - 18 - 2k + 10 + 3 = k - 5

So $|A| = 0$ when $k = 5$ .

Now check if $(\text{adj}A)B \neq 0$ at $k = 5$ to confirm no solution (not infinite solutions).

At $k = 5$ , substituting into the third equation: $5x + 9y + 5z = 4$ , but from equations 1 and 2, we can check the augmented matrix has inconsistency. The system is indeed inconsistent at $k = 5$ .

Answer: $k = 5$

For consistency questions, always do both steps: find when $|A| = 0$ , then check whether it’s infinite solutions or no solution. JEE sometimes gives options where both $k = 3$ and $k = 5$ appear — the trap is picking the wrong one.

PYQ 3 — JEE Main 2022

Question: If $A$ is a $3 \times 3$ matrix and $|A| = 5$ , find $|A \cdot \text{adj}(A)|$ .

Solution:

We know a beautiful identity: $A \cdot \text{adj}(A) = |A| \cdot I$

So:

|A \cdot \text{adj}(A)| = ||A| \cdot I| = |A|^n \cdot |I| = 5^3 \cdot 1 = 125

Answer: $125$

This is a 30-second question if you know the identity $A \cdot \text{adj}(A) = |A|I$ . It took students who didn’t know it 5+ minutes of painful cofactor computation — and they still got it wrong.

Difficulty Distribution

For JEE Main specifically:

Difficulty	Proportion	What It Looks Like
Easy (1–2 min)	~40%	Direct formula application — $
Medium (2–4 min)	~45%	3×3 determinant with parameter, consistency analysis, Cayley-Hamilton application
Hard (4+ min)	~15%	Matrix equations with unknowns, abstract properties, combined with other chapters

In JEE Advanced, matrices often appears as a paragraph-based question — 2–3 parts linked by one matrix setup. The difficulty jumps significantly. Focus on JEE Main pattern for 80% of your preparation; spend the last 20% on JEE Advanced PYQs from 2019–2024.

Expert Strategy

Week 1 — Build the formula bank. Don’t start with problems. Write all the determinant properties, adjoint formulas, and consistency conditions on one A4 sheet. Paste it where you study. The entire chapter runs on ~12 formulas.

Week 2 — Drill 3×3 determinant calculations. Speed here directly translates to marks. Practice expanding 3×3 determinants until you can do it in under 90 seconds. Use the “choose the row with most zeros” trick every time.

Week 3 — PYQs by topic. Do all JEE Main PYQs (2019–2024) grouped by sub-topic: determinant properties, inverse, system of equations, matrix algebra. You’ll see the same 5–6 question patterns repeat.

The 80/20 of this chapter: Master determinant properties + the adjoint/inverse formulas + consistency conditions, and you can solve roughly 80% of JEE Main questions from this chapter. These three areas appear together in almost every exam.

For JEE Advanced, practice questions involving:

Matrices satisfying polynomial equations (Cayley-Hamilton applications)
Systems with parameters where you need to analyse all three cases (unique, infinite, no solution)
Matrix functions and commutativity proofs

Common Traps

Trap 1 — Forgetting $|kA| = k^n|A|$ , not $k|A|$

If $|A| = 3$ and they ask $|2A|$ for a $3 \times 3$ matrix, the answer is $2^3 \times 3 = 24$ , not $6$ . This appears in JEE Main almost every year in some form.

Trap 2 — Matrix multiplication is NOT commutative

$AB \neq BA$ in general. When a question says “find $X$ such that $AX = B$ ”, you cannot write $X = B/A$ or $X = BA^{-1}$ . It must be $X = A^{-1}B$ . The order matters.

Trap 3 — Confusing “infinite solutions” and “no solution” when $|A| = 0$

Both cases require $|A| = 0$ . The difference is whether the augmented matrix has rank equal to the coefficient matrix. Students often mark “infinite solutions” without checking — and lose marks when the system is actually inconsistent.

Trap 4 — The adjoint of a 2×2 matrix swap

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , the adjoint is $\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ . Students often forget to swap $a$ and $d$ (they negate all elements instead). The diagonal elements swap; the off-diagonal elements negate.

Trap 5 — Using Cramer’s rule when $|A| = 0$

Cramer’s rule only works when $|A| \neq 0$ . If a question asks you to solve a system and $|A| = 0$ , switch to row reduction or consistency analysis immediately. Blindly applying Cramer’s gives division by zero.

One last thought on strategy: this chapter has the best effort-to-marks ratio in JEE Maths. A student who has done 50 focused PYQs from this chapter will almost certainly land 8 marks per paper here. That’s not something you can guarantee with Integration or Probability. Treat it as your banking topic — nail the fundamentals, protect those marks.