Paper I — Mathematics

Matrices and Determinants

Exam-focused notes, visual explanation, traps, revision sheet and every mapped past NDA question.

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High yield5 mapped NDA questions

Build the idea

Matrices and determinants are a dependable NDA scoring area. Questions usually test order and multiplication, determinant properties, inverse matrices, or a short system of linear equations. Work symbolically first: expanding a determinant too early often creates avoidable arithmetic.

| abcd |=ad − bc

A 2 × 2 determinant

Concept

A matrix is a rectangular array of numbers; a determinant is a scalar associated with a square matrix. Matrices are used to represent and solve systems of linear equations (Cramer's rule, matrix-inverse method).

Matrix multiplication is not commutative (AB ≠ BA in general).

Key Points & Formulas

Determinant of [[a,b],[c,d]] = ad − bc.
Triangular/diagonal matrix: determinant = product of diagonal entries.
A is singular (no inverse) ⇔ det A = 0.
Inverse: A⁻¹ = adj(A)/det(A); product order: (m×n)(n×p) → m×p.

Worked Example

Q. For what value of k is the matrix [[k, 3], [2, 4]] singular?

Singular ⇒ determinant = 0.

det = 4k − 6 = 0 ⇒ k = 6/4 = 3/2.

Common traps

Assuming AB = BA. Matrix multiplication is generally not commutative.
Using an inverse when det(A) = 0; a singular matrix has no inverse.
Swapping two rows without changing the determinant's sign.

NDA shortcuts

For a triangular matrix, multiply the diagonal entries.
Before multiplying, check only the inner dimensions; the outer dimensions give the answer's order.
Use determinant properties to create zeros before a 3×3 expansion.

One-screen revision sheet

Recall before you practise

det([[a,b],[c,d]]) = ad-bc
A⁻¹ = adj(A)/det(A)
det(AB) = det(A)det(B)

Past-paper bank

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5 NDA questions

Choose an answer before the solution appears. Option order is mixed automatically.

Question 1

NDA I 2026

If $M^T$ is the transpose of a $2 \times 2$ matrix $M$ , then which of the following is/are correct?
I. $|M + M^T| = |M| + |M^T|$ if $M$ is symmetric.
II. $|M + M^T| = 0$ if $M$ is anti-symmetric.
Select the answer using the code given below:

Question 2

NDA I 2025

If $\omega$ is a non-real cube root of unity, then what is a root of the following equation?

\begin{vmatrix} x+1 & \omega & \omega^2 \\ \omega & x+\omega^2 & 1 \\ \omega^2 & 1 & x+\omega \end{vmatrix} = 0

Question 3

NDA I 2024

If A = [[sin2θ, 2sin²θ − 1, 0], [cos2θ, 2sinθcosθ, 0], [0, 0, 1]], then which of the following statements is/are correct?
1. A⁻¹ = adj A
2. A is skew-symmetric matrix
3. A⁻¹ = Aᵀ

Question 4

NDA II 2024

Consider matrices P = [[0, c, −b], [−c, 0, a], [b, −a, 0]] and Q = [[a^2, ab, ac], [ab, b^2, bc], [ac, bc, c^2]]. Which of the following is/are correct? I. PQ is a null matrix. II. QP is an identity matrix of order 3. III. PQ = QP.

Question 5

NDA I 2024

For what value of n is the determinant | C(9,4) C(9,3) C(10,n−2); C(11,6) C(11,5) C(12,n); C(m,7) C(m,6) C(m+1,n+1) | = 0 for every m > n?

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