How much of JEE Main comes from Matrices and Determinants?

JEEGenie estimates about 5% of JEE Main marks come from Matrices and Determinants, based on recent papers. The share varies from year to year.

Which formulas matter most in Matrices and Determinants?

The Matrices and Determinants sheet on this page has 23 formula entries, starting with Matrix Definition and Order, Types of Matrices, Equality of Matrices, Matrix Addition and Scalar Multiplication, Matrix Multiplication. Each one comes with when to use it and the trap to avoid.

What topics does Matrices and Determinants cover for JEE?

Matrices and Determinants covers Types of Matrices, Matrix Operations, Determinants and Properties, Inverse of a Matrix, System of Linear Equations, Cramer's Rule.

How should I revise Matrices and Determinants before the exam?

Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Matrices and Determinants questions on JEEGenie.

How many previous-year questions are there on Matrices and Determinants?

JEEGenie has 635 previous-year questions on Matrices and Determinants, and this page shows a few with worked explanations.

Is practice on Matrices and Determinants free on JEEGenie?

Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.

Matrices and Determinants for JEE Main & Advanced — formulas, what is asked, and traps

JEEGenie estimates about 5% of JEE Main marks come from Matrices and Determinants. This page covers 23 formula entries, each with when to use it, what JEE asks and the traps. A common slip: Confusing rows and columns when writing order; always write rows first.

Key points to know

Matrix Definition and Order. A matrix is a rectangular array of numbers arranged in rows and columns. Its order is given by $m \times n$ , where $m$ is the number of rows and $n$ is the number of columns.
Types of Matrices. Special matrices include row, column, zero, square, diagonal, scalar, identity, upper/lower triangular, and symmetric/skew-symmetric matrices, each with distinct properties.
Equality of Matrices. Two matrices are equal if they have the same order and corresponding entries are equal.
Transpose of a Matrix. The transpose of a matrix $A$ is obtained by interchanging rows and columns, denoted $A^{T}$ .
Determinant of a Matrix. A determinant is a scalar value computed from a square matrix, indicating invertibility and volume scaling.
Adjoint of a Matrix. The adjoint of a square matrix $A$ is the transpose of its cofactor matrix, denoted $adj (A)$ .
Rank of a Matrix. The rank of a matrix is the maximum number of linearly independent rows or columns; it equals the order of the largest non-zero minor.
Symmetric and Skew-Symmetric Matrices. A matrix is symmetric if $A^{T} = A$ ; skew-symmetric if $A^{T} = - A$ . Diagonal entries of skew-symmetric are zero.
Minor and Cofactor. Minor $M_{i j}$ is determinant of submatrix after removing row $i$ and column $j$ . Cofactor $C_{i j} = (- 1)^{i + j} M_{i j}$ .
Singular and Non-Singular Matrices. A square matrix is singular if $\det (A) = 0$ (non-invertible); non-singular if $\det (A) \neq 0$ (invertible).
Orthogonal Matrix. A square matrix $A$ is orthogonal if $A^{T} A = A A^{T} = I$ , implying $A^{- 1} = A^{T}$ .
Characteristic Equation and Eigenvalues. For a square matrix $A$ , the characteristic equation is $\det (A - λ I) = 0$ , whose roots are eigenvalues $λ$ .

Formula sheet

Name	Formula	When to use
Matrix Definition and Order	$A = [a_{i j}]_{m \times n}$	When describing or identifying the size of any matrix.
Types of Matrices	$I_{n} = diag (1, 1, \dots, 1)$ ; $A^{T} = A$ for symmetric, $A^{T} = - A$ for skew-symmetric.	When classifying matrices or applying specific properties in problems.
Equality of Matrices	$A = B ⟺ a_{i j} = b_{i j}$ for all $i, j$ .	When solving for unknown entries in matrix equations.
Matrix Addition and Scalar Multiplication	$(A + B)_{i j} = a_{i j} + b_{i j}$ ; $(k A)_{i j} = k \cdot a_{i j}$	When combining matrices or scaling them in linear combinations.
Matrix Multiplication	$(A B)_{i j} = \sum_{k = 1}^{n} a_{i k} b_{k j}$	When combining linear transformations or solving systems of equations.
Transpose of a Matrix	$(A^{T})_{i j} = a_{j i}$	When dealing with symmetric properties or simplifying expressions.
Determinant of a Matrix	$\det (A)$ or $∣ A ∣$ ; for $2 \times 2$ : $∣ \begin{array}{cc} a & b \\ c & d \end{array} ∣ = a d - b c$	To check if a matrix is invertible (non-zero determinant) or to solve systems via Cramer's rule.
Properties of Determinants	$\det (A B) = \det (A) \det (B)$ ; $\det (k A) = k^{n} \det (A)$	When simplifying determinant expressions or proving identities.
Adjoint of a Matrix	$adj (A) = (C_{i j})^{T}$ where $C_{i j}$ is cofactor of $a_{i j}$	When finding inverse or solving matrix equations using $A^{- 1} = \frac{adj (A)}{\det (A)}$ .
Inverse of a Matrix	$A^{- 1} = \frac{1}{\det (A)} adj (A)$	To solve $A X = B$ as $X = A^{- 1} B$ , or to verify matrix properties.
Solving Linear Systems Using Matrices	$X = A^{- 1} B$ ; Cramer's rule: $x_{i} = \frac{\det (A_{i})}{\det (A)}$	When solving linear equations with same number of equations as unknowns.
Rank of a Matrix	$rank (A) = number of non-zero rows in row echelon form$	To determine consistency of linear systems or linear independence.
Elementary Row Operations	$R_{i} \leftrightarrow R_{j}$ , $R_{i} \to k R_{i}$ , $R_{i} \to R_{i} + k R_{j}$	When computing inverse via Gauss-Jordan or solving systems.
Symmetric and Skew-Symmetric Matrices	$A^{T} = A$ (symmetric); $A^{T} = - A$ (skew-symmetric)	When decomposing a matrix into symmetric and skew-symmetric parts: $A = \frac{A + A^{T}}{2} + \frac{A - A^{T}}{2}$ .
Cramer's Rule	$x_{i} = \frac{\det (A_{i})}{\det (A)}$	When solving small systems (2 or 3 variables) with unique solution.
Minor and Cofactor	$C_{i j} = (- 1)^{i + j} M_{i j}$	When computing determinant via expansion or adjoint.
Properties of Transpose	$(A B)^{T} = B^{T} A^{T}$	When simplifying expressions involving transposes.
Singular and Non-Singular Matrices	$\det (A) = 0 ⟹$ singular; $\det (A) \neq 0 ⟹$ non-singular	To check invertibility or existence of unique solutions.
Trace of a Matrix	$tr (A) = \sum_{i = 1}^{n} a_{i i}$	When dealing with eigenvalues or matrix invariants.
Orthogonal Matrix	$A^{T} A = I$	When dealing with rotations or preserving lengths.
Matrix Polynomials	$p (A) = a_{n} A^{n} + \dots + a_{0} I$ ; Cayley-Hamilton: if $p (λ) = \det (A - λ I)$ then $p (A) = O$	To compute high powers of matrices or find inverse using polynomial.
Characteristic Equation and Eigenvalues	$\det (A - λ I) = 0$	To find eigenvalues, which are used in diagonalization and stability analysis.
Cayley-Hamilton Theorem	$A^{n} + c_{n - 1} A^{n - 1} + \dots + c_{0} I = 0$	To compute $A^{- 1}$ or high powers of $A$ without diagonalization.

Open the Matrices and Determinants formula page

What JEE asks

Matrix Definition and Order. JEE often asks to determine the order from given entries or to compare matrices of different orders.
Types of Matrices. JEE frequently tests identification of matrix types from given conditions, e.g., 'If $A^{T} = - A$ , then $A$ is skew-symmetric'.
Equality of Matrices. JEE often sets up equations by equating corresponding elements of two matrices.
Matrix Addition and Scalar Multiplication. JEE tests these operations in the context of matrix equations and properties like distributivity.
Matrix Multiplication. JEE often asks to compute products or use properties like $(A B)^{T} = B^{T} A^{T}$ .
Transpose of a Matrix. JEE tests transpose properties in proofs and matrix equations.
Determinant of a Matrix. JEE frequently asks to compute determinants of $3 \times 3$ matrices using Sarrus or cofactor expansion.
Properties of Determinants. JEE uses these properties to evaluate determinants without direct expansion.
Adjoint of a Matrix. JEE tests adjoint properties like $A \cdot adj (A) = \det (A) I$ .
Inverse of a Matrix. JEE often asks to find inverse of $2 \times 2$ or $3 \times 3$ matrices, or use inverse to solve systems.
Solving Linear Systems Using Matrices. JEE tests solving systems using matrix methods, often with parameters.
Rank of a Matrix. JEE asks to find rank by reducing to echelon form or using minors.
Elementary Row Operations. JEE tests row operations to find inverse or rank.
Symmetric and Skew-Symmetric Matrices. JEE often asks to prove properties or find such decomposition.
Cramer's Rule. JEE uses Cramer's rule for systems with parameters, checking consistency.
Minor and Cofactor. JEE tests computation of minors and cofactors in determinant problems.
Properties of Transpose. JEE tests these in proofs and matrix equations.
Singular and Non-Singular Matrices. JEE often asks to find conditions for singularity (e.g., values of parameters).
Trace of a Matrix. JEE tests trace in properties like $tr (A + B) = tr (A) + tr (B)$ .
Orthogonal Matrix. JEE tests properties like rows/columns form orthonormal set.
Matrix Polynomials. JEE uses Cayley-Hamilton to find $A^{- 1}$ or $A^{n}$ .
Characteristic Equation and Eigenvalues. JEE asks to find eigenvalues for $2 \times 2$ matrices and use properties like sum = trace, product = determinant.
Cayley-Hamilton Theorem. JEE often asks to verify theorem for a given matrix or use it to find inverse.

Common traps

Confusing rows and columns when writing order; always write rows first. Matrix Definition and Order
Assuming all diagonal matrices are scalar; scalar matrices have all diagonal entries equal. Types of Matrices
Comparing matrices of different orders; they cannot be equal. Equality of Matrices
Adding matrices of different orders; they must be conformable. Matrix Addition and Scalar Multiplication
Multiplying in wrong order; matrix multiplication is not commutative ( $A B \neq B A$ generally). Matrix Multiplication
Confusing transpose with inverse; they are different. Transpose of a Matrix
Applying determinant formula to non-square matrices; only square matrices have determinants. Determinant of a Matrix
Assuming $\det (A + B) = \det (A) + \det (B)$ ; this is false. Properties of Determinants
Confusing adjoint with inverse; adjoint is used to compute inverse. Adjoint of a Matrix
Assuming every square matrix has an inverse; only non-singular matrices do. Inverse of a Matrix
Applying Cramer's rule when $\det (A) = 0$ ; system may be inconsistent or have infinite solutions. Solving Linear Systems Using Matrices
Confusing rank with determinant; a non-zero determinant implies full rank, but zero determinant doesn't mean rank 0. Rank of a Matrix
Using column operations when row operations are specified; they are different. Elementary Row Operations
Assuming a symmetric matrix must be diagonal; it only needs symmetry about diagonal. Symmetric and Skew-Symmetric Matrices

Topics in Matrices and Determinants

Types of Matrices
Matrix Operations
Determinants and Properties
Inverse of a Matrix
System of Linear Equations
Cramer's Rule

Previous-year questions on Matrices and Determinants

JEEGenie has 635 previous-year questions on Matrices and Determinants. A few worked examples follow.

Year	Questions
2026	33
2025	36
2024	47
2023	53
2022	48
2021	69
2020	33
2019	91
2018	16
2017	17
2016	21
2015	8
2014	14
2013	15
2012	10
2011	15
2010	13
2009	2
2008	5
2007	2
2006	2
2005	7
2004	12
2003	6
2002	2
2001	3
2000	3
1999	2
1998	1
1997	5
1995	1
1994	2
1993	3
1992	2
1991	1
1990	2
1989	2
1988	2
1987	2
1986	3
1985	2
1984	2
1983	5
1982	3
1981	6
1979	4
1978	2

Question 1 hard

Let

a \in R

and

A

be a matrix of order

3 \times 3

such that

\det (A) = - 4

and

A + I = [\begin{array}{lll} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{array}]

, where

I

is the identity matrix of order

3 \times 3

. If

\det ((a + 1) adj ((a - 1) A))

2^{m} 3^{n}, m

n \in {0, 1, 2, \dots, 20}

, then

m + n

is equal to :

Answer: D

Explanation. We have

A + I = [\begin{array}{ccc} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{array}]

, so

A = [\begin{array}{ccc} 0 & a & 1 \\ 2 & 0 & 0 \\ a & 1 & 1 \end{array}]

. Given

\det (A) = - 4

. Compute

\det (A)

\det (A) = 0 \cdot (0 \cdot 1 - 0 \cdot 1) - a \cdot (2 \cdot 1 - 0 \cdot a) + 1 \cdot (2 \cdot 1 - 0 \cdot a) = - a (2) + 1 (2) = - 2 a + 2.

Set

- 2 a + 2 = - 4 \Rightarrow - 2 a = - 6 \Rightarrow a = 3

. Now compute

\det ((a + 1) adj ((a - 1) A)) = \det (4 adj (2 A))

. For a

3 \times 3

matrix

B

adj (k B) = k^{2} adj (B)

and

\det (adj (B)) = (\det B)^{2}

. Here

B = 2 A

, so

\det (B) = 2^{3} \det (A) = 8 \cdot (- 4) = - 32

. Then

\det (adj (2 A)) = (- 32)^{2} = 1024 = 2^{10}

. Now

\det (4 adj (2 A)) = 4^{3} \cdot 2^{10} = 64 \cdot 2^{10} = 2^{6} \cdot 2^{10} = 2^{16}

. Thus

2^{16} = 2^{m} 3^{n}

gives

m = 16

n = 0

, so

m + n = 16

. The answer is option D.

From NTA/IIT papers: JEE Main 2025 · 2 Apr Shift 1

Frequently asked questions

How much of JEE Main comes from Matrices and Determinants?: JEEGenie estimates about 5% of JEE Main marks come from Matrices and Determinants, based on recent papers. The share varies from year to year.
Which formulas matter most in Matrices and Determinants?: The Matrices and Determinants sheet on this page has 23 formula entries, starting with Matrix Definition and Order, Types of Matrices, Equality of Matrices, Matrix Addition and Scalar Multiplication, Matrix Multiplication. Each one comes with when to use it and the trap to avoid.
What topics does Matrices and Determinants cover for JEE?: Matrices and Determinants covers Types of Matrices, Matrix Operations, Determinants and Properties, Inverse of a Matrix, System of Linear Equations, Cramer's Rule.
How should I revise Matrices and Determinants before the exam?: Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Matrices and Determinants questions on JEEGenie.
How many previous-year questions are there on Matrices and Determinants?: JEEGenie has 635 previous-year questions on Matrices and Determinants, and this page shows a few with worked explanations.
Is practice on Matrices and Determinants free on JEEGenie?: Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.