← Engineering Mathematics · DSSSB AE Civil

Chapter 10 of 11

Linear Algebra

In the DSSSB AE Civil syllabus under Engineering Mathematics · 2 parts

Matrices & Determinants · Eigenvalues & Eigenvectors

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📑 Contents (16 sections)

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

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

Part 2 of 2

Eigenvalues & Eigenvectors

Last reviewed 16 Sept 2026 · 7 min read

Definition

For a square matrix $A$ , a non-zero vector $X$ is an eigenvector and the scalar $λ$ the corresponding eigenvalue (characteristic root, latent root) if

A X = λ X

i.e. $A$ only stretches or shrinks $X$ without changing its direction.

Characteristic equation

(A - λ I) X = O has non-trivial solutions iff ∣ A - λ I ∣ = 0

$∣ A - λ I ∣ = 0$ is the characteristic equation, a polynomial of degree $n$ with $n$ eigenvalues (counting multiplicity, possibly complex).
For a $2 \times 2$ matrix: $λ^{2} - (trace) λ + ∣ A ∣ = 0$ .
For a $3 \times 3$ matrix: $λ^{3} - (trace) λ^{2} + (sum of principal minors of order 2) λ - ∣ A ∣ = 0$ .

Eigenvectors for each $λ$ are found by solving $(A - λ I) X = O$ ; any non-zero multiple of an eigenvector is also an eigenvector.

Properties of eigenvalues

Property	Statement
Sum	$\sum λ_{i} =$ trace of $A$ (sum of diagonal elements)
Product	$\prod λ_{i} = ∣ A ∣$ → $A$ is singular iff at least one eigenvalue is zero
Triangular/diagonal matrices	Eigenvalues are the diagonal elements
Transpose	$A$ and $A^{T}$ have the same eigenvalues
Scalar multiple	$k A$ has eigenvalues $k λ$
Powers	$A^{m}$ has eigenvalues $λ^{m}$ (same eigenvectors)
Inverse	$A^{- 1}$ has eigenvalues $1/ λ$ (same eigenvectors)
Polynomial	$p (A)$ has eigenvalues $p (λ)$ , e.g. $A^{2} + 2 A + I$ → $λ^{2} + 2 λ + 1$
Shift	$A - k I$ has eigenvalues $λ - k$
Adjoint	$adj A$ has eigenvalues $∣ A ∣/ λ$
Similar matrices ( $B = P^{- 1} A P$ )	Same eigenvalues (and characteristic polynomial)
Distinct eigenvalues	Corresponding eigenvectors are linearly independent

Special matrices

Matrix	Eigenvalues	Eigenvectors
Real symmetric	All real	Eigenvectors of distinct eigenvalues are orthogonal; always diagonalisable
Skew-symmetric (real)	Zero or purely imaginary	—
Orthogonal	Unit modulus ( $∣ λ ∣ = 1$ ); if $λ$ is an eigenvalue, so is $1/ λ$
Hermitian	Real
Idempotent	0 or 1
Nilpotent	All zero
Involutory	±1

Algebraic and geometric multiplicity

Algebraic multiplicity (AM) — number of times $λ$ repeats as a root.
Geometric multiplicity (GM) — number of linearly independent eigenvectors for $λ$ $= n - ρ (A - λ I)$ .
$1 \leq GM \leq A M$ ; a matrix is diagonalisable iff GM = AM for every eigenvalue.

Cayley–Hamilton theorem

Every square matrix satisfies its own characteristic equation.

If $∣ A - λ I ∣ = λ^{n} + c_{1} λ^{n - 1} + \dots + c_{n} = 0$ , then

A^{n} + c_{1} A^{n - 1} + \dots + c_{n} I = O

Uses:

Inverse: multiply by $A^{- 1}$ and rearrange (for non-singular $A$ ).
Higher powers of $A$ expressed in terms of lower powers.

Diagonalisation

If $A$ ( $n \times n$ ) has $n$ linearly independent eigenvectors, form the modal matrix $P$ with eigenvectors as columns:

P^{- 1} A P = D = diag (λ_{1}, λ_{2}, \dots, λ_{n})

Then $A^{m} = P D^{m} P^{- 1}$ — useful for powers.
For a real symmetric matrix, $P$ can be made orthogonal (normalised eigenvectors) → $P^{T} A P = D$ (orthogonal transformation).

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