Mathematics · JEE

Matrices and Determinants Formula Sheet for JEE

62+ JEE formulas in this unit

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The Matrices and Determinants JEE formula sheet lists 62+ important formulas for JEE Main and Advanced, including essential identities from Matrices, algebra of matrices, type of matrices, Determinants and matrices of order two and three, Evaluation of determinants, Area of triangles using determinants, and more. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Matrices and Determinants. This unit-wise formula list covers 62+ exam-relevant results across Matrices, algebra of matrices, type of matrices, Determinants and matrices of order two and three, Evaluation of determinants, Area of triangles using determinants, and more, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

62 formulas across 6 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 41Important: 19Supplementary: 2

Matrices, algebra of matrices, type of matrices

Order of a matrixEssential

A=[a_{ij}]_{m\times n}

has order

m\times n

with

m

rows and

n

columns.

Variables: $a_{ij}$ : entry in the $i$ -th row and $j$ -th column.
Conditions: Applicable to any matrix.
Where used in JEE: Identifying dimensions, checking possibility of operations.

Equality of matricesEssential

A=B\iff

both have the same order and

a_{ij}=b_{ij}

for all

i,j

Variables: $A=[a_{ij}],\ B=[b_{ij}]$ .
Conditions: Matrices must be of the same order.
Where used in JEE: Finding unknown entries, solving matrix equations.

Matrix addition and subtractionEssential

(A\pm B)_{ij}=a_{ij}\pm b_{ij}

Variables: $A=[a_{ij}],\ B=[b_{ij}]$ .
Conditions: $A$ and $B$ must have the same order.
Where used in JEE: Basic matrix algebra, simplification of expressions.

Scalar multiplicationEssential

(kA)_{ij}=k a_{ij}

Variables: $k$ : scalar, $A=[a_{ij}]$ .
Conditions: Applicable to any matrix.
Where used in JEE: Linear combinations of matrices, solving equations.

Matrix multiplication definitionEssential

A

is of order

m\times n

and

B

is of order

n\times p

, then

AB

is of order

m\times p

and

(AB)_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}

Variables: $A=[a_{ik}],\ B=[b_{kj}]$ .
Conditions: Number of columns of $A$ equals number of rows of $B$ .
Where used in JEE: Product evaluation, matrix equations, linear systems.

Row by column rule for 2x2 multiplicationImportant

\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}e&f\\g&h\end{bmatrix}=\begin{bmatrix}ae+bg&af+bh\\ce+dg&cf+dh\end{bmatrix}

Variables: $a,b,c,d,e,f,g,h$ : matrix entries.
Conditions: Both matrices are of order $2\times 2$ .
Where used in JEE: Direct computation of matrix products in JEE problems.

Non-commutativity of matrix multiplicationEssential

In general,

AB\ne BA

Variables: $A,B$ : matrices of compatible orders.
Conditions: Both products must be defined when compared.
Where used in JEE: Counterexamples, simplification cautions, solving equations.

Associative law of multiplicationEssential

(AB)C=A(BC)

Variables: $A,B,C$ : matrices of compatible orders.
Conditions: All products involved must be defined.
Where used in JEE: Rearranging products, powers of matrices.

Distributive lawsEssential

A(B+C)=AB+AC

(A+B)C=AC+BC

Variables: $A,B,C$ : matrices of compatible orders.
Conditions: The sums and products involved must be defined.
Where used in JEE: Expansion and simplification of matrix expressions.

Scalar with matrix productImportant

k(AB)=(kA)B=A(kB)

Variables: $k$ : scalar, $A,B$ : compatible matrices.
Conditions: $AB$ must be defined.
Where used in JEE: Factorization and simplification.

Zero matrix product caveatImportant

AB=O

does not necessarily imply

A=O

B=O

Variables: $O$ : zero matrix.
Conditions: Compatible dimensions.
Where used in JEE: True/false statements, conceptual questions.

Transpose definitionEssential

A=[a_{ij}]

, then

A^T=[a_{ji}]

Variables: $A^T$ : transpose of $A$ .
Conditions: Applicable to any matrix.
Where used in JEE: Symmetric/skew-symmetric matrices, product transpose.

Transpose of sum and scalar multipleImportant

(A+B)^T=A^T+B^T

(kA)^T=kA^T

Variables: $A,B$ : same order, $k$ : scalar.
Conditions: $A+B$ must be defined.
Where used in JEE: Simplification involving transpose.

Transpose of productEssential

(AB)^T=B^T A^T

Variables: $A,B$ : compatible matrices.
Conditions: $AB$ must be defined.
Where used in JEE: Manipulating matrix products, proving symmetry.

Double transposeSupplementary

(A^T)^T=A

Variables: $A$ : any matrix.
Where used in JEE: Simplification of transpose expressions.

Symmetric matrix conditionEssential

A

is symmetric

\iff A^T=A

Variables: $A$ : square matrix.
Conditions: Defined only for square matrices.
Where used in JEE: Classification of matrices, decomposition questions.

Skew-symmetric matrix conditionEssential

A

is skew-symmetric

\iff A^T=-A

Variables: $A$ : square matrix.
Conditions: Defined only for square matrices.
Where used in JEE: Classification and decomposition of matrices.

Diagonal entries of skew-symmetric matrixImportant

For skew-symmetric

A

a_{ii}=0

for all

i

Variables: $a_{ii}$ : diagonal entries of $A$ .
Conditions: Over real numbers or any field of characteristic not equal to 2.
Where used in JEE: Forming/sketching skew-symmetric matrices.

Decomposition into symmetric and skew-symmetric partsImportant

A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)

Variables: First term is symmetric, second term is skew-symmetric.
Conditions: Applicable to any square matrix over characteristic not equal to 2.
Where used in JEE: Expressing a matrix as sum of symmetric and skew-symmetric matrices.

Identity matrix propertyEssential

For identity matrix

I_n

AI_n=I_mA=A

whenever orders are compatible.

Variables: $I_n$ : identity matrix of order $n$ .
Conditions: If $A$ is $m\times n$ , then $AI_n$ and $I_mA$ are defined.
Where used in JEE: Matrix equations, inverses, simplification.

Powers of a square matrixImportant

A^n=A\cdot A\cdots A

(

n

factors), with

A^0=I

Variables: $A$ : square matrix, $n\in\mathbb{N}\cup\{0\}$ .
Conditions: Defined only for square matrices.
Where used in JEE: Recurrence relations, algebraic identities in matrices.

Types of matricesImportant

Row matrix: order

1\times n

; column matrix: order

m\times 1

; square matrix: order

n\times n

; rectangular matrix:

m\times n,\ m\ne n

; zero matrix: all entries

0

; diagonal matrix:

a_{ij}=0

for

i\ne j

; scalar matrix:

a_{ij}=0

for

i\ne j

a_{11}=a_{22}=\cdots=a_{nn}=k

; identity matrix: scalar matrix with

k=1

; upper triangular:

a_{ij}=0

for

i>j

; lower triangular:

a_{ij}=0

for

i<j

Variables: $a_{ij}$ : entries of the matrix.
Conditions: As stated for each type.
Where used in JEE: Recognition and use of structural properties.

Determinants and matrices of order two and three

Determinant of a 2x2 matrixEssential

\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc

Variables: $a,b,c,d$ : entries of the matrix.
Conditions: Applicable to matrices of order $2\times 2$ .
Where used in JEE: Inverse, area, solving linear equations, expansion.

Determinant of a 3x3 matrix by expansionEssential

\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)

Variables: $a,b,c,d,e,f,g,h,i$ : entries of the matrix.
Conditions: Applicable to matrices of order $3\times 3$ .
Where used in JEE: Direct determinant calculation in JEE problems.

Sarrus rule for 3x3 determinantImportant

\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}=aei+bfg+cdh-ceg-bdi-afh

Variables: $a,b,c,d,e,f,g,h,i$ : entries of the matrix.
Conditions: Applicable only for $3\times 3$ determinants.
Where used in JEE: Quick evaluation of determinants of order 3.

Minor of an elementEssential

The minor of

a_{ij}

M_{ij}=

determinant obtained by deleting the

i

-th row and

j

-th column.

Variables: $M_{ij}$ : minor of $a_{ij}$ .
Conditions: For a square determinant.
Where used in JEE: Cofactors, adjoint, Laplace expansion.

Cofactor of an elementEssential

C_{ij}=(-1)^{i+j}M_{ij}

Variables: $C_{ij}$ : cofactor, $M_{ij}$ : minor.
Conditions: For a square determinant.
Where used in JEE: Expansion of determinant, adjoint, inverse.

Evaluation of determinants

Laplace expansion along a row or columnEssential

\det(A)=\sum_{j=1}^{n} a_{ij}C_{ij}=\sum_{i=1}^{n} a_{ij}C_{ij}

Variables: $C_{ij}$ : cofactor of $a_{ij}$ .
Conditions: For any square matrix $A$ .
Where used in JEE: Evaluation by suitable row/column expansion.

Sign pattern of cofactors for 3x3Important

\begin{bmatrix}+&-&+\\-&+&-\\+&-&+\end{bmatrix}

Variables: Signs correspond to $(-1)^{i+j}$ .
Conditions: Useful for cofactor expansion.
Where used in JEE: Avoiding sign errors in determinant and adjoint problems.

Determinant of identity and diagonal matricesEssential

\det(I_n)=1

, and for diagonal or triangular matrix

A

\det(A)=\prod_{i=1}^{n} a_{ii}

Variables: $a_{ii}$ : diagonal entries.
Conditions: For square diagonal, upper triangular, or lower triangular matrices.
Where used in JEE: Fast determinant evaluation.

Determinant unchanged by transposeImportant

\det(A^T)=\det(A)

Variables: $A$ : square matrix.
Where used in JEE: Simplification and property-based evaluation.

Interchange of two rows or two columnsEssential

If two rows or two columns are interchanged, determinant changes sign.

Conditions: For any determinant.
Where used in JEE: Row/column operation based evaluation.

Two equal rows or columnsEssential

If any two rows or any two columns are identical, then

\det(A)=0

Conditions: For any square matrix.
Where used in JEE: Testing singularity, simplifying determinants.

Proportional rows or columnsEssential

If one row/column is a scalar multiple of another, then

\det(A)=0

Conditions: For any square matrix.
Where used in JEE: Detecting zero determinant quickly.

Common factor from a row or columnEssential

If every entry of a row or column has common factor

k

, then

\det(A)=k\det(B)

, where

B

is obtained by factoring out

k

Variables: $k$ : scalar.
Where used in JEE: Simplifying determinant values.

Determinant with a row or column of zerosEssential

If any row or any column is entirely zero, then

\det(A)=0

Where used in JEE: Immediate evaluation.

Row/column replacement operationEssential

Adding to one row (or column) a scalar multiple of another row (or column) leaves the determinant unchanged.

Conditions: Operation must be of the form $R_i\to R_i+kR_j$ or $C_i\to C_i+kC_j$ , $i\ne j$ .
Where used in JEE: Evaluation of determinants by reduction.

Scaling a row or columnImportant

If one row or column is multiplied by

k

, the determinant is multiplied by

k

Variables: $k$ : scalar.
Where used in JEE: Tracking determinant change under operations.

Determinant of a productEssential

\det(AB)=\det(A)\det(B)

Variables: $A,B$ : square matrices of same order.
Conditions: Both must be square of same order.
Where used in JEE: Inverse, product determinants, theoretical questions.

Determinant of scalar multipleImportant

For

A

of order

n\times n

\det(kA)=k^n\det(A)

Variables: $k$ : scalar, $n$ : order of square matrix.
Where used in JEE: Parameter-based determinant problems.

Determinant power propertyImportant

\det(A^m)=(\det A)^m

Variables: $A$ : square matrix, $m\in\mathbb{N}$ .
Where used in JEE: Powers of matrices and determinant relations.

Singular and non-singular matrix testEssential

A

is singular

\iff \det(A)=0

;

A

is non-singular

\iff \det(A)\ne 0

Variables: $A$ : square matrix.
Where used in JEE: Existence of inverse, consistency and uniqueness tests.

Area of triangles using determinants

Area of triangle from coordinatesEssential

\text{Area}=\frac{1}{2}\left|\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}\right|=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|

Variables: $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ : vertices of triangle.
Conditions: Points lie in a plane.
Where used in JEE: Coordinate geometry and determinant-based area questions.

Condition for collinearity using determinantEssential

\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}=0\iff \text{the three points are collinear}

Variables: $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ : three points.
Conditions: In the plane.
Where used in JEE: Testing collinearity, line geometry.

Adjoint and inverse of a square matrix

Adjoint definitionEssential

\operatorname{adj}(A)=(C_{ij})^T

, where

C_{ij}

is the cofactor of

a_{ij}

Variables: $\operatorname{adj}(A)$ : adjoint (adjugate) of $A$ .
Conditions: Defined for square matrices.
Where used in JEE: Finding inverse by adjoint method.

Adjoint of a 2x2 matrixEssential

\operatorname{adj}\begin{bmatrix}a&b\\c&d\end{bmatrix}=\begin{bmatrix}d&-b\\-c&a\end{bmatrix}

Variables: $a,b,c,d$ : entries of the matrix.
Conditions: For a $2\times 2$ matrix.
Where used in JEE: Quick inverse of order 2 matrices.

Fundamental adjoint relationEssential

A\,\operatorname{adj}(A)=\operatorname{adj}(A)A=\det(A)I

Variables: $I$ : identity matrix of same order as $A$ .
Conditions: $A$ must be square.
Where used in JEE: Deriving inverse, matrix identities.

Inverse by adjoint formulaEssential

A^{-1}=\dfrac{1}{\det(A)}\operatorname{adj}(A)

Variables: $A^{-1}$ : inverse of $A$ .
Conditions: Exists only if $\det(A)\ne 0$ .
Where used in JEE: Finding inverse of 2x2 and 3x3 matrices.

Inverse existence criterionEssential

A^{-1}

exists

\iff \det(A)\ne 0

Variables: $A$ : square matrix.
Where used in JEE: Checking invertibility before solving matrix equations.

Inverse identitiesEssential

AA^{-1}=A^{-1}A=I

(A^{-1})^{-1}=A

(AB)^{-1}=B^{-1}A^{-1}

Variables: $A,B$ : invertible square matrices.
Conditions: Required inverses must exist.
Where used in JEE: Simplifying matrix equations, product inverses.

Inverse of transposeImportant

(A^T)^{-1}=(A^{-1})^T

Variables: $A$ : invertible square matrix.
Conditions: $A^{-1}$ must exist.
Where used in JEE: Problems involving transpose and inverse together.

Determinant of inverseImportant

\det(A^{-1})=\dfrac{1}{\det(A)}

Variables: $A$ : invertible square matrix.
Conditions: $\det(A)\ne 0$ .
Where used in JEE: Determinant-inverse relation questions.

Inverse of a 2x2 matrixEssential

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

Variables: $a,b,c,d$ : entries of the matrix.
Conditions: $ad-bc\ne 0$ .
Where used in JEE: Direct inverse calculation, solving two-variable linear systems.

Inverse of scalar multipleSupplementary

(kA)^{-1}=\dfrac{1}{k}A^{-1}

Variables: $k$ : nonzero scalar, $A$ : invertible matrix.
Conditions: $k\ne 0$ and $A^{-1}$ exists.
Where used in JEE: Simplification of inverse expressions.

Test of consistency and solution of simultaneous linear equations in two or three variables using matrices

Linear system in matrix formEssential

A system of linear equations can be written as

AX=B

Variables: $A$ : coefficient matrix, $X$ : column matrix of variables, $B$ : constant column matrix.
Conditions: Dimensions must be compatible.
Where used in JEE: Solving simultaneous linear equations using matrices.

Solution by inverse matrix methodEssential

AX=B

and

A^{-1}

exists, then

X=A^{-1}B

Variables: $A$ : coefficient matrix, $X$ : variable column matrix, $B$ : constant column matrix.
Conditions: $A$ must be square and non-singular.
Where used in JEE: Solving systems with unique solution.

Cramer's rule for two variablesImportant

For

a_1x+b_1y+c_1=0,\ a_2x+b_2y+c_2=0

, with

D=\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix}

D_x=\begin{vmatrix}-c_1&b_1\\-c_2&b_2\end{vmatrix}

D_y=\begin{vmatrix}a_1&-c_1\\a_2&-c_2\end{vmatrix}

, then

x=\dfrac{D_x}{D},\ y=\dfrac{D_y}{D}

Variables: $D$ : determinant of coefficients.
Conditions: $D\ne 0$ .
Where used in JEE: Direct solution of 2-variable linear equations.

Cramer's rule for three variablesImportant

For

AX=B

, if

D=\det(A)\ne 0

, then

x=\dfrac{D_x}{D},\ y=\dfrac{D_y}{D},\ z=\dfrac{D_z}{D}

, where

D_x,D_y,D_z

are obtained by replacing the corresponding coefficient columns of

A

by the constants column.

Variables: $D$ : determinant of coefficient matrix.
Conditions: Applicable to a square system with $D\ne 0$ .
Where used in JEE: Direct solution of 3-variable linear equations.

Consistency criterion for two variablesEssential

For

a_1x+b_1y+c_1=0

and

a_2x+b_2y+c_2=0

: unique solution if

\dfrac{a_1}{a_2}\ne\dfrac{b_1}{b_2}

; no solution if

\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\ne\dfrac{c_1}{c_2}

; infinitely many solutions if

\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}

Variables: $a_i,b_i,c_i$ : coefficients and constants.
Conditions: Denominators considered only when defined; equivalent determinant form may also be used.
Where used in JEE: Testing consistency of pair of linear equations.

Consistency criterion using determinants for two variablesEssential

Unique solution if

D\ne 0

; no solution if

D=0

and at least one of

D_x,D_y\ne 0

; infinitely many solutions if

D=D_x=D_y=0

Variables: $D$ : coefficient determinant, $D_x,D_y$ : Cramer determinants.
Conditions: For a 2 $\times$ 2 linear system.
Where used in JEE: Determinant-based test of consistency.

Consistency criterion for three variables via determinantEssential

For a 3-variable system

AX=B

: unique solution if

\det(A)\ne 0

; if

\det(A)=0

, then the system may be either inconsistent or have infinitely many solutions.

Variables: $A$ : coefficient matrix.
Conditions: For square systems of order 3.
Where used in JEE: Quick test before applying inverse/Cramer method.

Homogeneous system criterionImportant

For homogeneous system

AX=O

: trivial solution always exists; non-trivial solution exists if

\det(A)=0

. If

\det(A)\ne 0

, only trivial solution exists.

Variables: $O$ : zero column matrix.
Conditions: For square homogeneous linear systems.
Where used in JEE: Existence of non-zero solutions in parameter problems.

Frequently asked questions

What are the important Matrices and Determinants formulas for JEE?

This page lists 62+ JEE-relevant Matrices and Determinants formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Matrices and Determinants formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Matrices and Determinants, covering Matrices, algebra of matrices, type of matrices, Determinants and matrices of order two and three, Evaluation of determinants, Area of triangles using determinants, and more.

How should I revise the Matrices and Determinants formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Matrices and Determinants MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Matrices and Determinants?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.

JEE Formula Sheet

Matrices, algebra of matrices, type of matrices

Determinants and matrices of order two and three

Evaluation of determinants

Area of triangles using determinants

Adjoint and inverse of a square matrix

Test of consistency and solution of simultaneous linear equations in two or three variables using matrices

Popular questions in Matrices and Determinants

Frequently asked questions

What are the important Matrices and Determinants formulas for JEE?

Is this Matrices and Determinants formula sheet aligned with JEE Main?

How should I revise the Matrices and Determinants formula sheet before JEE?

Where can I practise Matrices and Determinants MCQs after revising formulas?

Does this replace NCERT for Matrices and Determinants?

Related topics