NDA Maths · Teaching notes

Matrices & Determinants — NDA Mathematics

Matrices & Determinants is the single biggest scoring chapter in NDA Mathematics — 170 past-year questions across 2017–2026, around eight or nine marks on every paper. It is also the hardest: nearly a third of the questions are HARD, and two areas (determinant properties and special determinants) sit near 50% HARD. The chapter is almost entirely a small set of rules applied carefully, so the payoff is in knowing the properties cold and not falling for the standard traps. Work the six notes below in order — matrices and their algebra, the special types, determinant evaluation and properties, the special determinants, the adjoint–inverse machinery, and finally linear systems — and the bank turns into rule-application.

NDA Maths strategy NDA home

Subtopic notes

PYQ weightage by concept

36 concepts · 170 PYQs — where the marks actually sit, so you know what to drill first

Matrices: Order, Algebra, Powers & Equations33 PYQs · 19%

Concept	PYQs	Share
Matrix multiplication and conformability	13	8%
Matrix polynomials and equations	7	4%
Powers of a matrix	4	2%
Matrix algebra — where numbers' rules break	4	2%
Counting matrices	3	2%
Equality, addition, and scalar multiplication	1	1%
Transpose and its rules	1	1%
What a matrix is — order and typesfoundation	—	—

Special Matrices and Their Tell-Tale Properties23 PYQs · 14%

Concept	PYQs	Share
Symmetric and skew-symmetric matrices	8	5%
The special-matrix catalog	5	3%
Orthogonal matrices	5	3%
Rotation matrices	3	2%
Diagonal, scalar, and identity matrices	1	1%
Idempotent and involutory matrices	1	1%

Determinants: Evaluation & Properties58 PYQs · 34%

Concept	PYQs	Share
Determinant of products, scalars, and powers	18	11%
Singular matrices and determinant equations	8	5%
Factor-theorem and Vandermonde determinants	7	4%
Structured and bounded determinants	7	4%
Core row and column properties	6	4%
Cyclic determinants	5	3%
Sums and sequences of determinants	4	2%
Evaluating 2×2 and 3×3 determinants	3	2%

Special Determinants: Trig, Complex, ω, Polynomial19 PYQs · 11%

Concept	PYQs	Share
Trigonometric determinants	7	4%
Polynomial and progression determinants	6	4%
Determinants with complex entries	3	2%
Cube-root-of-unity determinants	3	2%

Cofactors, Adjoint & Inverse29 PYQs · 17%

Concept	PYQs	Share
Adjoint power formulas	7	4%
The adjoint (adjugate)	6	4%
Inverse properties and the reversal law	6	4%
Minors, cofactors, and expansion	4	2%
Inverse via the adjoint	4	2%
Inverses of special matrices	2	1%

Linear Systems: Consistency & Cramer's Rule8 PYQs · 5%

Concept	PYQs	Share
Consistency from the coefficient determinant	3	2%
Cramer's rule	2	1%
Finding the parameter for consistency	2	1%
Homogeneous systems and the solution space	1	1%

Formula & revision sheet

19 formulas · 1 reference tables · 2 gotchas across all subtopics — the exam-eve cheat-sheet

Matrices: Order, Algebra, Powers & Equations

Formulas (3)

Matrix multiplication and conformability · Order of a product $A_{m\times n}\, B_{n\times p} = (AB)_{m\times p}$
Transpose and its rules · Transpose of a product (reversal law) $(AB)^T = B^T A^T$
Matrix polynomials and equations · Cayley–Hamilton (2×2) $A^2 - (a+d)\,A + (ad - bc)\,I = O$

Watch out for (1)

Don't import $a^2 - b^2 = (a+b)(a-b)$ into matrices
→ Matrix algebra — where numbers' rules break

Special Matrices and Their Tell-Tale Properties

Formulas (4)

Symmetric and skew-symmetric matrices · Symmetric + skew decomposition $A = \underbrace{\tfrac12(A + A^T)}_{\text{symmetric}} + \underbrace{\tfrac12(A - A^T)}_{\text{skew-symmetric}}$
Diagonal, scalar, and identity matrices · Diagonal determinant and inverse $\det D = \prod_i d_i, \qquad D^{-1} = \operatorname{diag}\!\left(\tfrac{1}{d_1}, \dots, \tfrac{1}{d_n}\right)$
Orthogonal matrices · Orthogonality $AA^T = I \;\Longrightarrow\; A^{-1} = A^T,\quad \det A = \pm 1$
Rotation matrices · Rotation composition $R(\theta)\,R(\phi) = R(\theta + \phi), \qquad R(\theta)^n = R(n\theta)$

Reference tables (1)

The special-matrix catalog6 rows

Type	Defining property	Key consequence
Symmetric	$A^T = A$	$a_{ij} = a_{ji}$ ; inverse (if any) is symmetric
Skew-symmetric	$A^T = -A$	diagonal all 0; odd order $\Rightarrow \det = 0$ Odd-order skew-symmetric is ALWAYS singular (det 0). Even-order need not be.
Diagonal	off-diagonal all 0	$\det =$ product of diagonal entries
Orthogonal	$AA^T = I$	$A^{-1} = A^T$ ; $\det = \pm 1$
Idempotent	$A^2 = A$	$\det \in \{0, 1\}$
Involutory	$A^2 = I$	$A^{-1} = A$ ; $\det = \pm 1$

Recognise the defining equation first; the determinant and inverse follow immediately.

Determinants: Evaluation & Properties

Formulas (4)

Evaluating 2×2 and 3×3 determinants · 2×2 determinant $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$
Determinant of products, scalars, and powers · Multiplicativity and scaling $\det(AB) = \det A\,\det B, \qquad \det(kA) = k^{\,n}\det A$
Factor-theorem and Vandermonde determinants · Vandermonde (3×3) $\begin{vmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{vmatrix} = (a-b)(b-c)(c-a)$
Cyclic determinants · Cyclic determinant $\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix} = -(a^3+b^3+c^3-3abc)$

Watch out for (1)

$\det(kA) = k^n \det A$ , NOT $k\det A$
→ Determinant of products, scalars, and powers

Special Determinants: Trig, Complex, ω, Polynomial

Formulas (2)

Determinants with complex entries · Powers of i (period 4) $i = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1$
Cube-root-of-unity determinants · Cube roots of unity $\omega^3 = 1, \qquad 1 + \omega + \omega^2 = 0$

Cofactors, Adjoint & Inverse

Formulas (5)

Minors, cofactors, and expansion · Cofactor expansion vs alien cofactors $\sum_j a_{ij}C_{ij} = \det A, \qquad \sum_j a_{ij}C_{kj} = 0\ (k \neq i)$
The adjoint (adjugate) · Adjoint identity $A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|\,I_n$
Adjoint power formulas · Adjoint of an n×n matrix $|\operatorname{adj}A| = |A|^{\,n-1}, \qquad \operatorname{adj}(\operatorname{adj}A) = |A|^{\,n-2}A$
Inverse via the adjoint · Inverse formula $A^{-1} = \frac{1}{|A|}\operatorname{adj}A \quad (|A| \neq 0)$
Inverse properties and the reversal law · Reversal law for inverses $(AB)^{-1} = B^{-1}A^{-1}$

Linear Systems: Consistency & Cramer's Rule

Formulas (1)

Cramer's rule · Cramer's rule $x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}$