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Math · Algebra

Matrices formulas for JEE

Every Matrices formula you need for JEE, grouped by concept.

44 formulas5 concepts
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Number of ElementsN=mnN = mnTotal number of elements in a matrix of order m by n.matrixelementsorder
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All 44 Matrices formulas
01

Matrices Basics

4 formulas

Number of Elements

N=mnN = mn

Total number of elements in a matrix of order m by n.

matrixelementsorder

Equality of Matrices

A=B  ⟺  aij=bijA = B \iff a_{ij} = b_{ij}

Condition for two matrices to be exactly equal.

applies whenMatrices A and B must be of the exact same order.
equalityelements

Order of a Matrix

A=[aij]m×nA = [a_{ij}]_{m \times n}

Notation for a matrix with m rows and n columns.

applies when1≤i≤m,1≤j≤n,i,j∈N1 \le i \le m, 1 \le j \le n, i,j \in \mathbb{N}
matrixordernotation

Trace of a Matrix

Tr(A)=∑i=1naii\text{Tr}(A) = \sum_{i=1}^{n} a_{ii}

Sum of the principal diagonal elements.

applies whenA must be a square matrix.
tracediagonaljee-advanced
02

Matrix Operations

20 formulas

Associative Law for Addition

(A+B)+C=A+(B+C)(A + B) + C = A + (B + C)

Grouping does not affect matrix addition.

applies whenMatrices must be of the same order.
additionassociativeproperties

Commutative Law for Addition

A+B=B+AA + B = B + A

Matrix addition is independent of order.

applies whenMatrices must be of the same order.
additioncommutativeproperties

Matrix Addition

C=A+B  ⟹  cij=aij+bijC = A + B \implies c_{ij} = a_{ij} + b_{ij}

Element-wise addition of two matrices.

applies whenMatrices A and B must have the exact same order.
additionoperations

Difference of Matrices

A−B=A+(−1)B=[aij−bij]A - B = A + (-1)B = [a_{ij} - b_{ij}]

Subtracting one matrix from another.

applies whenMatrices A and B must be of the same order.
subtractiondifference

Function Matrix Additive Property

F(x)F(y)=F(x+y)F(x)F(y) = F(x+y)

Product of specific trigonometric function matrices sums their arguments.

applies whenWhere F(x)=[cos⁡x−sin⁡x0sin⁡xcos⁡x0001]F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}
function-matrixtrigonometricadditive

Idempotent Matrix

A2=AA^2 = A

A matrix which, when multiplied by itself, yields itself.

applies whenA must be a square matrix.
idempotentpowerjee-advanced

Involutory Matrix

A2=I  ⟹  A−1=AA^2 = I \implies A^{-1} = A

A matrix that is its own inverse.

applies whenA must be a square matrix.
involutorypowerinversejee-advanced

Matrix Multiplication Element

cik=∑j=1naijbjkc_{ik} = \sum_{j=1}^{n} a_{ij}b_{jk}

The (i, k)-th element of product matrix C=AB.

applies whenNumber of columns in A must equal number of rows in B.
multiplicationproduct

Associative Law for Multiplication

A(BC)=(AB)CA(BC) = (AB)C

Matrix multiplication grouping can be shifted.

applies whenMatrix orders must allow both sides to be defined.
multiplicationassociativeproperties

Distributive Law for Multiplication

A(B+C)=AB+ACA(B + C) = AB + AC

Matrix multiplication distributes over matrix addition.

applies whenMatrix orders must be compatible for addition and multiplication.
multiplicationdistributiveproperties

Multiplicative Identity

IA=AI=AIA = AI = A

Multiplying by Identity matrix returns the same matrix.

applies whenA must be a square matrix.
identitymultiplication

Negative of a Matrix

−A=(−1)A=[−aij]-A = (-1)A = [-a_{ij}]

Finding the additive inverse of a matrix.

negativeadditive_inverse

Nilpotent Matrix

Ak=OA^k = O

A square matrix such that some power of it is the zero matrix.

applies whenA must be a square matrix, k is a positive integer.
nilpotentpowerzero-matrixjee-advanced

N-th Power of Rotation Matrix

A=[cos⁡θsin⁡θ−sin⁡θcos⁡θ]  ⟹  An=[cos⁡nθsin⁡nθ−sin⁡nθcos⁡nθ]A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \implies A^n = \begin{bmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{bmatrix}

Identity for taking the power of a standard 2D rotation matrix.

applies whenn is a natural number (from PMI application).
rotationpowertrigonometric

Scalar Distributive Law (Over Matrix Add)

k(A+B)=kA+kBk(A + B) = kA + kB

Scalar multiplies across added matrices.

scalardistributiveproperties

Scalar Distributive Law (Over Scalar Add)

(k+l)A=kA+lA(k + l)A = kA + lA

Matrix multiplies across added scalars.

scalardistributiveproperties

Scalar Multiplication

kA=[kaij]kA = [k a_{ij}]

Multiplication of a matrix by a scalar constant.

applies whenk is any scalar constant.
scalarmultiplication

Tangent Half-Angle Matrix Identity

(I+A)=(I−A)[cos⁡α−sin⁡αsin⁡αcos⁡α](I + A) = (I - A)\begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}

Expression of rotation matrix in terms of tangent half-angle matrix.

applies whenWhere A=[0−tan⁡(α/2)tan⁡(α/2)0]A = \begin{bmatrix} 0 & -\tan(\alpha/2) \\ \tan(\alpha/2) & 0 \end{bmatrix}
tangenthalf-anglerotation

Linearity of Trace

Tr(kA+B)=kTr(A)+Tr(B)\text{Tr}(kA + B) = k\text{Tr}(A) + \text{Tr}(B)

Trace is a linear operator over matrix addition and scalar multiplication.

applies whenA and B are square matrices of the same order, k is scalar.
tracelinearityjee-advanced

Trace of Matrix Product

Tr(AB)=Tr(BA)\text{Tr}(AB) = \text{Tr}(BA)

Trace is invariant under cyclic permutations of matrix products.

applies whenAB and BA must both be defined square matrices.
traceproductcyclicjee-advanced
03

Transpose of a Matrix

15 formulas

Symmetry of Transformed Matrix

(B′AB)′=B′A′B=±B′AB(B'AB)' = B'A'B = \pm B'AB

Matrix B'AB takes on the symmetry properties of A.

applies whenSign depends on if A is symmetric (+) or skew-symmetric (-).
symmetricskew-symmetrictransform

Orthogonal Matrix

AAT=ATA=I  ⟹  A−1=ATAA^T = A^TA = I \implies A^{-1} = A^T

A square matrix whose transpose equals its inverse.

applies whenA must be a square matrix.
orthogonaltransposeinversejee-advanced

Determinant of Skew-Symmetric Matrix

∣A∣=0|A| = 0

Odd order skew-symmetric matrices have zero determinant.

applies whenA is a skew-symmetric matrix of odd order.
determinantskew-symmetricoddjee-advanced

Diagonal of Skew-Symmetric Matrix

aii=0a_{ii} = 0

All diagonal elements of a skew-symmetric matrix are strictly zero.

applies whenA is a skew-symmetric square matrix.
skew-symmetricdiagonalzero

Skew-Symmetric Matrix Definition

A′=−A  ⟹  aij=−ajiA' = -A \implies a_{ij} = -a_{ji}

A matrix equal to the negative of its transpose.

applies whenA must be a square matrix.
skew-symmetrictranspose

Matrix Splitting Theorem

A=12(A+A′)+12(A−A′)A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A')

Expressing any square matrix as sum of a symmetric and skew-symmetric matrix.

applies whenA must be a square matrix.
splittingsymmetricskew-symmetric

Symmetric/Skew Matrix Generators

B=A+A′ (Sym),C=A−A′ (Skew-sym)B = A + A' \text{ (Sym)}, C = A - A' \text{ (Skew-sym)}

Generating symmetric and skew-symmetric matrices from any square matrix A.

applies whenA must be a square matrix with real entries.
symmetricskew-symmetricgenerator

Commutativity condition for Symmetric AB

(AB)′=AB  ⟺  AB=BA(AB)' = AB \iff AB = BA

Product of symmetric matrices is symmetric if and only if they commute.

applies whenA and B are symmetric matrices of the same order.
symmetriccommutative

Skew-Symmetry of Commutator

(AB−BA)′=−(AB−BA)(AB - BA)' = -(AB - BA)

Difference of cross-products of symmetric matrices is skew-symmetric.

applies whenA and B are both symmetric matrices of the same order.
symmetricskew-symmetriccommutator

Symmetric Matrix Definition

A′=A  ⟹  aij=ajiA' = A \implies a_{ij} = a_{ji}

A matrix equal to its transpose.

applies whenA must be a square matrix.
symmetrictranspose

Transpose of a Matrix

A′=[aji]n×mA' = [a_{ji}]_{n \times m}

Interchanging rows and columns.

applies whenA is an m by n matrix.
transposedefinition

Reversal Law for Transpose

(AB)′=B′A′(AB)' = B'A'

Transpose of a product reverses the order of multiplication.

applies whenAB must be defined.
transposemultiplicationreversal

Transpose of Scalar Multiple

(kA)′=kA′(kA)' = kA'

Scalars can be pulled out of transpose.

transposescalarproperties

Transpose of Sum

(A+B)′=A′+B′(A + B)' = A' + B'

Transpose distributes over matrix addition.

applies whenA and B must be of the same order.
transposeadditionproperties

Transpose of Transpose

(A′)′=A(A')' = A

Taking the transpose twice returns original matrix.

transposeproperties
04

Invertible Matrices

3 formulas

Inverse using Adjoint

A−1=1∣A∣adj(A)A^{-1} = \frac{1}{|A|}\text{adj}(A)

Formula for finding the inverse using the adjugate matrix.

applies when∣A∣≠0|A| \neq 0 (Matrix A must be non-singular).
inverseadjointdeterminantjee-advanced

Invertible Matrix Definition

AB=BA=I  ⟹  B=A−1AB = BA = I \implies B = A^{-1}

Existence of an inverse matrix B for a given matrix A.

applies whenA and B must be square matrices of the same order.
inverseinvertibleidentity

Reversal Law for Inverse

(AB)−1=B−1A−1(AB)^{-1} = B^{-1}A^{-1}

Inverse of a product reverses the order of inversion.

applies whenA and B must be invertible square matrices of the same order.
inversemultiplicationreversal
05

Non-commutativity of Matrices

2 formulas

Matrix Non-Commutativity

AB≠BAAB \neq BA

In general, matrix multiplication does not commute.

applies whenApplies generally, even if A and B are square matrices.
non-commutativemultiplication

Zero Product Property Failure

AB=O̸  ⟹  A=O or B=OAB = O \not\implies A = O \text{ or } B = O

Product of two non-zero matrices can be a zero matrix.

zero-matrixnullproductfallacy
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