← Engineering Mathematics · OPSC Assistant Agriculture Engineer

Chapter 3 of 3

Algebra, Matrices & Determinants

In the OPSC Assistant Agriculture Engineer syllabus under Engineering Mathematics · 2 parts

Algebra — Complex Numbers, Quadratic Equations, Progressions & Binomial Theorem · Matrices & Determinants

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📑 Contents (16 sections)

Part 1 of 2

Algebra — Complex Numbers, Quadratic Equations, Progressions & Binomial Theorem

Last reviewed 16 Sept 2026 · 7 min read

Complex numbers

A complex number $z = x + i y$ , where $i = - 1$ ( $i^{2} = - 1$ , $i^{3} = - i$ , $i^{4} = 1$ ); $x = Re (z)$ , $y = Im (z)$ .

Forms

Form	Expression
Algebraic (Cartesian)	$z = x + i y$
Polar (trigonometric)	$z = r (cos θ + i sin θ)$
Exponential (Euler)	$z = r e^{i θ}$

Modulus $∣ z ∣ = r = x^{2} + y^{2}$ .
Argument $θ = ar g z$ with $tan θ = y / x$ , chosen in the correct quadrant; principal argument in $(- π, π]$ .
Conjugate $\overset{z}{ˉ} = x - i y$ ; $z \overset{z}{ˉ} = ∣ z ∣^{2}$ .
Euler's formula: $e^{i θ} = cos θ + i sin θ$ ; $e^{iπ} + 1 = 0$ .

Properties

$∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣∣ z_{2} ∣$ ; $ar g (z_{1} z_{2}) = ar g z_{1} + ar g z_{2}$ .
$\frac{z _{1}}{z _{2}} = \frac{∣ z _{1} ∣}{∣ z _{2} ∣}$ ; $ar g \frac{z _{1}}{z _{2}} = ar g z_{1} - ar g z_{2}$ .
Triangle inequality: $∣ z_{1} + z_{2} ∣ \leq ∣ z_{1} ∣ + ∣ z_{2} ∣$ .
Division: multiply numerator and denominator by the conjugate of the denominator.

De Moivre's theorem

(cos θ + i sin θ)^{n} = cos n θ + i sin n θ

nth roots

The $n$ distinct $n$ th roots of $z = r e^{i θ}$ :

z_{k} = r^{1/ n} [cos \frac{θ + 2 k π}{n} + i sin \frac{θ + 2 k π}{n}], k = 0, 1, \dots, n - 1

They lie on a circle of radius $r^{1/ n}$ , equally spaced by $2 π / n$ .

Cube roots of unity

$1, ω = \frac{- 1 + i 3}{2}, ω^{2} = \frac{- 1 - i 3}{2}$

$ω^{3} = 1$ ; $1 + ω + ω^{2} = 0$ ; $\overset{ω}{ˉ} = ω^{2}$ .
Sum of all $n$ th roots of unity = 0; product = $(- 1)^{n - 1}$ .

Quadratic equations

a x^{2} + b x + c = 0, x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Nature of roots (real coefficients)

Discriminant $D = b^{2} - 4 a c$	Roots
$D > 0$	Real and distinct (rational if $D$ is a perfect square and coefficients rational)
$D = 0$	Real and equal ( $- b /2 a$ )
$D < 0$	Complex conjugates

Relations between roots and coefficients

Sum $α + β = - \frac{b}{a}$ ; product $α β = \frac{c}{a}$ .
$α^{2} + β^{2} = (α + β)^{2} - 2 α β$ ; $(α - β)^{2} = (α + β)^{2} - 4 α β$ ; $α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)$ .
Equation with roots $α, β$ : $x^{2} - (α + β) x + α β = 0$ .
For a cubic $a x^{3} + b x^{2} + c x + d = 0$ : $\sum α = - b / a$ , $\sum α β = c / a$ , $α β γ = - d / a$ .
Quadratic $a x^{2} + b x + c$ has a minimum (if $a > 0$ ) or maximum (if $a < 0$ ) at $x = - b /2 a$ , value $- D /4 a$ .

Progressions

FormulaArithmetic progression (AP)

Terms $a, a + d, a + 2 d, \dots$

T_{n} = a + (n - 1) d, S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)

Arithmetic mean of $a$ and $b$ : $A = \frac{a + b}{2}$

FormulaGeometric progression (GP)

Terms $a, a r, a r^{2}, \dots$

T_{n} = a r^{n - 1}, S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1} (r \neq = 1)

Infinite GP ( $∣ r ∣ < 1$ ): $S_{\infty} = \frac{a}{1 - r}$ Geometric mean of $a$ and $b$ (positive): $G = ab$

FormulaHarmonic progression (HP)

Reciprocals form an AP. Harmonic mean of $a$ and $b$ : $H = \frac{2 ab}{a + b}$ For positive numbers: $A \geq G \geq H$ and $G^{2} = A H$ .

Useful sums: $\sum n = \frac{n ( n + 1 )}{2}$ ; $\sum n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$ ; $\sum n^{3} = [\frac{n ( n + 1 )}{2}]^{2}$ .

Part 2 of 2

Matrices & Determinants

Last reviewed 16 Sept 2026 · 7 min read

Matrices

A matrix is a rectangular array of numbers arranged in m rows and n columns (order $m \times n$ ).

Types of matrices

Type	Definition
Row / column matrix	Single row / single column
Square matrix	$m = n$
Diagonal matrix	Square; all off-diagonal elements zero
Scalar matrix	Diagonal with all diagonal elements equal ( $k I$ )
Identity (unit) matrix $I$	Diagonal elements 1, others 0
Null (zero) matrix	All elements zero
Upper / lower triangular	All elements below / above the diagonal are zero
Symmetric	$A^{T} = A$ ( $a_{ij} = a_{j i}$ )
Skew-symmetric	$A^{T} = - A$ — diagonal elements are zero
Orthogonal	$A A^{T} = A^{T} A = I$ → $A^{- 1} = A^{T}$ , $∣ A ∣ = \pm 1$
Idempotent	$A^{2} = A$
Nilpotent	$A^{k} = O$ for some positive integer $k$
Involutory	$A^{2} = I$ ( $A^{- 1} = A$ )
Singular / non-singular	$∣ A ∣ = 0$ / $∣ A ∣ \neq = 0$
Hermitian	$A^{θ} = A$ (conjugate transpose equals itself)
Skew-Hermitian	$A^{θ} = - A$
Unitary	$A A^{θ} = I$

Every square matrix can be written uniquely as the sum of a symmetric and a skew-symmetric matrix:

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

Matrix operations

Addition — same order, element-wise; commutative and associative.
Scalar multiplication — multiply every element.
Multiplication $A B$ — defined when columns of A = rows of B; $(A B)_{ij} = \sum_{k} a_{ik} b_{k j}$ ; not commutative in general ( $A B \neq = B A$ ); associative; distributive.
$A B = O$ does not imply $A = O$ or $B = O$ .

Properties of transpose

$(A^{T})^{T} = A$ ; $(A + B)^{T} = A^{T} + B^{T}$ ; $(k A)^{T} = k A^{T}$ ; $(A B)^{T} = B^{T} A^{T}$ .

Determinants

For a $2 \times 2$ matrix: $a c b d = a d - b c$ .

For a $3 \times 3$ matrix, expand along any row or column using cofactors:

∣ A ∣ = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}, C_{ij} = (- 1)^{i + j} M_{ij}

( $M_{ij}$ = minor — determinant after deleting row $i$ and column $j$ .)

Properties of determinants

$∣ A^{T} ∣ = ∣ A ∣$ .
Interchanging two rows (or columns) changes the sign.
Two identical (or proportional) rows → determinant zero.
Multiplying a row by $k$ multiplies the determinant by $k$ ; hence $∣ k A ∣ = k^{n} ∣ A ∣$ for an $n \times n$ matrix.
Adding a multiple of one row to another does not change the determinant.
$∣ A B ∣ = ∣ A ∣∣ B ∣$ .
Determinant of a triangular or diagonal matrix = product of diagonal elements.
Sum of products of elements of a row with cofactors of another row = 0.

Adjoint and inverse

Adjoint $adj A$ = transpose of the cofactor matrix.
$A (adj A) = (adj A) A = ∣ A ∣ I$ .

FormulaInverse and related identities (A of order n)

A^{- 1} = \frac{adj A}{∣ A ∣}, ∣ A ∣ \neq = 0

For $2 \times 2$ : $[a c b d]^{- 1} = \frac{1}{a d - b c} [d - c - b a]$

$∣ A^{- 1} ∣ = \frac{1}{∣ A ∣}$
$∣ adj A ∣ = ∣ A ∣^{n - 1}$
$adj (adj A) = ∣ A ∣^{n - 2} A$
$(A B)^{- 1} = B^{- 1} A^{- 1}$ ; $(A^{T})^{- 1} = (A^{- 1})^{T}$
$adj (A B) = adj (B) adj (A)$

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