Empty relation is the relation R in X given by R = φ ⊂ X × X.

Universal relation is the relation R in X given by R = X × X.

Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X.

Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.

Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R

Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.

Equivalence class [a] containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a.

A function f : X → Y is one-one (or injective) if

f (x₁) = f (x₂2) ⇒ x1 = x2 ∀ x1, x2 ∈ X.

A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f (x) = y.

A function f : X → Y is one-one and onto (or bijective), if f is both one-one and onto.

The composition of functions f : A → B and g : B → C is the function gof : A → C given by gof (x) = g(f (x)) ∀ x ∈ A

A function f : X → Y is invertible if ∃ g : Y → X such that gof = I_X and fog = I_Y

A function f : X → Y is invertible if and only if f is one-one and onto.

Given a finite set X, a function f : X → X is one-one (respectively onto) if and only if f is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for infinite set

A binary operation ∗ on a set A is a function ∗ from A × A to A.

n element e ∈ X is the identity element for binary operation ∗ : X × X → X, if a ∗ e = a = e ∗ a ∀ a ∈ X.

An element a ∈ X is invertible for binary operation ∗ : X × X → X, if there exists b ∈ X such that a ∗ b = e = b ∗ a where, e is the identity for the binary operation ∗. The element b is called inverse of a and is denoted by a^-1

An operation ∗ on X is commutative if a ∗ b = b ∗ a ∀ a, b in X.

An operation ∗ on X is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) ∀ a, b, c in X

The domains and ranges (principal value branches) of inverse trigonometric functions are given in the following table:

sin^-1x should not be confused with (sin x)^-1. In fact, (sin x)^-1 = $\frac{1}{sin x}$ and similarly for other trigonometric functions.

The value of an inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric functions.

y = sin^-1 x ⇒ x = sin y

x = sin y ⇒ y = sin^-1 x

sin (sin^-1 x) = x

sin ^-1(sin x) = x

sin^-1 $\frac{1}{x}$ = cosec^-1 x

cos^-1 (-x) = π - cos^-1 (x)

cos^-1 $\frac{1}{x}$ = sec^-1 x

cot^-1 (-x) = π - cot^-1 (x)

tan^-1 $\frac{1}{x}$ = cot^-1 x

sec^-1 (-x) = π - sec^-1 (x)

sin^-1 (-x) = - sin^-1x

tan^-1 (-x) = - tan^-1x

tan^-1 (x) + cot^-1 (x) = $\frac{\pi}{2}$

cosec^-1 (-x) = - cosec^-1x

sin^-1 (x) + cos^-1 (x) = $\frac{\pi}{2}$

cosec^-1 (x) + sec^-1 (x) = $\frac{\pi}{2}$

tan^-1 (x) + tan^-1 (y) = tan^-1 $\frac{x+y}{1-xy}$

2tan^-1 (x) = tan^-1 $\frac{2x}{1-x^2}$

tan^-1 (x) - tan^-1 (y) = tan^-1 $\frac{x-y}{1+xy}$

2tan^-1 (x) = sin^-1 $\frac{2x}{1+x^2}$ = cos^-1 $\frac{2x}{1+x^2}$

A matrix is an ordered rectangular array of numbers or functions.

A matrix having m rows and n columns is called a matrix of order m × n.

[A_ij] _{m * 1} is a column matrix.

[A_ij] _{1 * n} is a row matrix.

An m × n matrix is a square matrix if m = n.

A = [A_ij] _{m * m} is a diagonal matrix if A_ij = 0 when i ≠ j

A = [a_ij]_{n * n} is a scalar matrix if a _ij = 0, when i ≠ j, a = k, (k is some constant), when i = j.

A = [a_ij]_{n * n} is an identity matrix, if a_ij = 1, when i = j, a_ij = 0, when i ≠ j.

A zero matrix has all its elements as zero.

A = [a_ij] = [b_ij] = B if (i) A and B are of same order, (ii) [a_ij] = [b_ij] for all possible values of i and j.

kA = k[a_ij]_{m * n} = [k(a_ij)]_{m * n}

– A = (–1)A

A – B = A + (–1) B

A + B = B + A

(A + B) + C = A + (B + C), where A, B and C are of same order.

k(A + B) = kA + kB, where A and B are of same order, k is constant.

(k + l ) A = kA + lA, where k and l are constant.

A = [a_ij]_{m * n} and B = [b_jk]_{n * p} then C = AB = [c_ik]_{m * p} where [c_ik] = $\sum_{j=1}^{n} a_{ij} b_{jk}$

(i) A(BC) = (AB)C, (ii) A(B + C) = AB + AC, (iii) (A + B)C = AC + BC

If A = [a_ij]_{m × n}, then A′ or A^T = [a _ji ]_{n × m}

(i) (A′)′ = A, (ii) (kA)′ = kA′, (iii) (A + B)′ = A′ + B′, (iv) (AB)′ = B′A′

A is a symmetric matrix if A′ = A.

A is a skew symmetric matrix if A′ = – A.

Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix.

If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A and is denoted by A^-1 and A is the inverse of B.

Inverse of a square matrix, if it exists, is unique.

Elementary operations of a matrix are as follows:
(i) R_i ↔ R_j OR C_i ↔ C_j
(ii) R_i ↔ kR_i OR C_i ↔ kC_i
(iii) R_i ↔ R_i + kR_j OR C_i ↔ C_i + kC_j

Determinant of a matrix A = $[a_{11}]_{1*1}$ is given by $|a_{11}| = a_{11}$

Determinant of a matrix A = $\begin{bmatrix}a_{11} & a_{12} \\a_{21} & a_{22} \end{bmatrix}$ is given by |A| = $a_{11} * a_{22} - a_{12} * a_{21}$

For any square matrix A, the |A| satisfy following properties:

Area of a triangle with vertices (x_i, y_i) , (x_j, y_j) and (x_k, y_k) is given by

Minor of an element a_ij of the determinant of matrix A is the determinant obtained by deleting i ^th row and j ^th column and denoted by M_ij

Cofactor of $a_{ij}$ is given by $A_{ij}$ = $(-1)^{i+j}M_{ij}$

Value of determinant of a matrix A is obtained by sum of product of elements of a row (or a column) with corresponding cofactors. For example, $|A| = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13} $

If elements of one row (or column) are multiplied with cofactors of elements of any other row (or column), then their sum is zero. For example, $|A| = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13} = 0$

In below image $A_{ij}$ is co-factor of $a_{ij}$

A (adj A) = (adj A) A = | A| I, where A is square matrix of order n

A square matrix A is said to be singular or non-singular according as | A| = 0 or | A| ≠ 0.

If AB = BA = I, where B is square matrix, then B is called inverse of A. Also A ^-1 = B or B ^-1 = A and hence (A ^-1 ) ^-1 = A.

A square matrix A has inverse if and only if A is non-singular.

A^-1 = $\frac{1}{|A|} (adj A)$

Unique solution of equation AX = B is given by X = A ^-1 B, where |A| ≠ 0

A system of equation is consistent or inconsistent according as its solution exists or not.

For a square matrix A in matrix equation AX = B