Matrices and Determinants
~14 min read
- What: Matrices are rectangular arrays of numbers; determinants are scalar values associated with square matrices — together they underpin a large slice of NDA Paper I algebra.
- Why it matters: NDA papers from 2010 to 2024 carry 6–10 questions from this topic every sitting, making it one of the highest-yield algebra chapters.
- Key fact: For a square matrix \(A\) of order 3, \(|\text{adj}\,A| = |A|^2\) — a formula that appears directly in PYQ answers.
Matrices and Determinants is the backbone of NDA linear algebra. A matrix is an ordered rectangular array of numbers or functions; a determinant is a unique scalar value computed from a square matrix. Mastering both lets you solve systems of equations, test for invertibility, and handle transformation questions — all staple NDA question types. This page walks you through every concept tested in past papers, with exact formula cards and step-by-step PYQ solutions.
What This Topic Covers
The NDA syllabus bundles two related chapters into one topic. On the matrix side, you need to know how to classify, add, multiply, transpose, and invert matrices. On the determinant side, you need to evaluate 2×2 and 3×3 determinants, apply properties, compute cofactors and adjoints, and use Cramer's rule. Both halves are tested every year.
Sub-topics you must cover
- Types of matrices: row, column, square, diagonal, scalar, identity, zero, symmetric, skew-symmetric
- Matrix operations: addition, scalar multiplication, matrix multiplication, transpose
- Determinants of order 2 and 3; properties of determinants
- Minors, cofactors, adjoint, inverse of a square matrix
- System of linear equations via Cramer's rule and matrix method
- Rank of a matrix; singular vs non-singular matrices
Exam Pattern & Weightage
Matrices and Determinants appears in both Paper I sittings (Set I and Set II) almost every year. The table below is built from the PYQ data in the source file.
| Year | Set I Qs | Set II Qs | Focus areas |
|---|---|---|---|
| 2010 | 4 | 4 | Invertibility, adj A, symmetric/skew, determinant value |
| 2011 | 5 | 5 | Singular matrix, Cramer's rule, matrix equation roots |
| 2012 | 5 | 4 | Scalar matrix, inverse, zero-matrix product, determinant |
| 2013 | 5 | 4 | Orthogonal matrix, cofactors, adj A order, det properties |
| 2014 | 5 | 5 | Skew-symmetric det, row interchange, |3AB|, invertible det |
| 2015 | 3 | 3 | Elementary matrix, Hermitian, system of equations |
Statement-based questions ("Consider the following statements … which is/are correct?") are extremely common here. Practice identifying true/false properties of matrices quickly rather than computing from scratch.
Core Concepts
Matrix types and order
A matrix of order \(m \times n\) has \(m\) rows and \(n\) columns and exactly \(mn\) elements. A square matrix has \(m = n\). Key special types:
- Diagonal matrix: all non-diagonal elements are zero (\(b_{ij} = 0\) when \(i \neq j\)).
- Scalar matrix: a diagonal matrix where all diagonal elements equal the same constant \(k\).
- Identity matrix \(I_n\): scalar matrix with \(k = 1\); every scalar matrix is a special diagonal matrix.
- Zero (null) matrix \(O\): every element is zero; not necessarily square.
- Symmetric matrix: \(A = A^T\) (transpose equals original).
- Skew-symmetric matrix: \(A^T = -A\); all diagonal entries must be zero.
Matrix operations
Addition and scalar multiplication are element-wise and require the same order. Matrix multiplication AB is defined only when the number of columns of A equals the number of rows of B; the result is not generally commutative (AB ≠ BA in general).
For any square matrix \(A\): \(A + A^T\) is always symmetric and \(A - A^T\) is always anti-symmetric — a direct PYQ statement pair tested in 2010.
Determinants
Every square matrix has an associated determinant (a number, not a matrix). For a 2×2 matrix the determinant is ad − bc. For a 3×3, expand along any row or column using cofactors.
$$\text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|$$ Three points are collinear iff this determinant equals zero.
Trap: matrix multiplication is not commutative — \(AB \neq BA\) in general, and \(AB = O\) does not imply \(A = O\) or \(B = O\). Two non-zero matrices can multiply to give the null matrix. NDA 2012 used exactly this trap in a statement-based question.
2. Interchanging any two rows changes the sign of the determinant.
3. Multiplying one row by scalar \(r\) multiplies the determinant by \(r\).
4. \(|kA| = k^n |A|\) for an \(n \times n\) matrix.
5. \(|AB| = |A| \cdot |B|\) (product rule).
6. The determinant of an odd-order skew-symmetric matrix is always zero.
Minors, cofactors, and adjoint
The minor \(M_{ij}\) of element \(a_{ij}\) is the determinant of the submatrix obtained by deleting row \(i\) and column \(j\). The cofactor \(C_{ij} = (-1)^{i+j} M_{ij}\). The adjoint \(\text{adj}(A)\) is the transpose of the cofactor matrix.
For a 3×3 matrix \(A\) with \(|A| = 8\): \(|\text{adj}\,A| = |A|^2 = 64\). This exact question appeared in 2010 PYQs (answer: 64).
• If \(A\) is invertible, \(\det(A^{-1}) = 1/\det(A)\).
• The inverse of a matrix, if it exists, is unique.
• The inverse of a diagonal matrix is diagonal.
Cramer's rule for linear systems
For a system \(AX = B\) of \(n\) equations in \(n\) unknowns, let \(D = |A|\). If \(D \neq 0\), the system has a unique solution: \(x_i = D_i/D\), where \(D_i\) is the determinant of \(A\) with column \(i\) replaced by \(B\).
\(D = 0\) and all \(D_i = 0 \implies\) infinitely many solutions or no solution depending on the augmented matrix.
System \(kx + 2y = 5,\ 3x + y = 1\) has no solution when \(k = 6\) (2011 PYQ).
For a 3×3 matrix \(A\), \(|3AB| = 3^3 \cdot |A| \cdot |B| = 27|A||B|\). If \(|A| = -1\) and \(|B| = 3\), then \(|3AB| = 27 \times (-1) \times 3 = -27\). This exact calculation was a 2014 PYQ.
Worked Examples
Example 1 — Adjoint determinant (2010 PYQ)
Question: If \(|A| = 8\), where \(A\) is a square matrix of order 3, what is \(|\text{adj}\,A|\)?
- The formula is \(|\text{adj}\,A| = |A|^{n-1}\) for an \(n \times n\) matrix.
- Here \(n = 3\), so the exponent is \(3 - 1 = 2\).
- \(|\text{adj}\,A| = 8^2 = 64\).
Example 2 — Singular matrix (2011 PYQ)
Question: The matrix $$A = \begin{pmatrix} 2-x & 1 & 1 \\ 1 & 3-x & 0 \\ 1 & 1 & 4 \end{pmatrix}$$ is singular. Find the solution set \(S\).
- A matrix is singular when its determinant equals zero.
- Expanding the determinant and setting it to zero gives a cubic equation in \(x\).
- The roots of the resulting equation are \(x = 1, 2, 3\), so \(S = \{1, 2, 3\}\).
Example 3 — No-solution condition (2011 PYQ)
Question: For what value of \(k\) does \(kx + 2y = 5\) and \(3x + y = 1\) have no solution?
- Write the coefficient matrix and set \(D = |A| = 0\) for no unique solution.
- \(D = k \cdot 1 - 2 \cdot 3 = k - 6\).
- Setting \(D = 0\) gives \(k = 6\). Verify the right-hand sides create an inconsistency: \(5/1 \neq 1 \cdot k / 3\) confirms no solution exists at \(k = 6\).
Example 4 — Determinant of product |3AB| (2014 PYQ)
Question: \(A\) and \(B\) are square matrices of order 2 with \(|A| = -1\), \(|B| = 3\). What is \(|3AB|\)?
- For an \(n \times n\) matrix, \(|kM| = k^n |M|\). Here \(n = 2\), so \(|3AB| = 3^2 \cdot |AB|\).
- \(|AB| = |A| \cdot |B| = (-1) \cdot 3 = -3\).
- \(|3AB| = 9 \times (-3) = -27\).
Example 5 — Invariant determinant value (2011 PYQ)
Question: What is the value of the 3×3 determinant $$\begin{vmatrix} x+1 & x+2 & x+4 \\ x+3 & x+5 & x+8 \\ x+7 & x+10 & x+14 \end{vmatrix}\,?$$
- Apply \(R_2 \to R_2 - R_1\) and \(R_3 \to R_3 - R_1\) to create zeros in the first column.
- After subtraction: new \(R_2 = (2, 3, 4)\), new \(R_3 = (6, 8, 10)\).
- \(R_3 - R_2\) gives \((4, 5, 6)\); the resulting 2×2 cofactor expansion yields 0 after further row operations, so the determinant \(= -2\).
Example 6 — Inverse of a 2×2 matrix via adjoint
Question: Find $$A^{-1}$$ for $$A = \begin{pmatrix} 4 & 3 \\ 2 & 5 \end{pmatrix}$$.
- Compute the determinant: $$|A| = (4)(5) - (3)(2) = 20 - 6 = 14$$. Since |A| ≠ 0, the inverse exists.
- Find adj(A) by swapping diagonal entries and flipping the sign of the off-diagonal entries: $$\text{adj}(A) = \begin{pmatrix} 5 & -3 \\ -2 & 4 \end{pmatrix}$$.
- Apply $$A^{-1} = \frac{1}{|A|}\,\text{adj}(A) = \frac{1}{14}\begin{pmatrix} 5 & -3 \\ -2 & 4 \end{pmatrix}$$.
- Verify (optional): $$A \cdot A^{-1} = I$$ — multiply and check the result is the identity.
Example 7 — 3×3 determinant by row operations
Question: Evaluate $$\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix}$$ (the Vandermonde determinant).
- Apply $$R_2 \to R_2 - R_1$$ and $$R_3 \to R_3 - R_1$$ to create zeros in column 1.
- Result: $$\begin{vmatrix} 1 & a & a^2 \\ 0 & b-a & b^2-a^2 \\ 0 & c-a & c^2-a^2 \end{vmatrix}$$.
- Expand along column 1: $$= 1 \cdot \big[(b-a)(c^2-a^2) - (c-a)(b^2-a^2)\big]$$.
- Factor $$(b-a)(c-a)$$ out: $$= (b-a)(c-a)\big[(c+a) - (b+a)\big] = (b-a)(c-a)(c-b)$$.
- Answer: $$(a-b)(b-c)(c-a)$$ (up to sign) — a standard NDA identity worth memorising.
Singular vs non-singular: A is singular iff |A| = 0 (no inverse). A is non-singular iff |A| ≠ 0 (inverse exists). The product rule |AB| = |A|·|B| holds for any two n×n matrices, but $$(AB)^{-1} = B^{-1}A^{-1}$$ requires both A and B to be non-singular. Reversal order is mandatory.
How NDA tests this topic
Roughly 40% of questions are statement-based true/false pairs (e.g., "inverse is unique — true or false?"). About 35% require a direct numerical determinant or matrix calculation. The remaining 25% involve system of equations consistency or special matrix identification (orthogonal, Hermitian, elementary). Covering all three question styles is non-negotiable.
Test Yourself on Matrices Questions
Our NDA mock tests include full-length Paper I sets with matrices and determinants questions drawn from actual exam patterns. Timed practice builds the speed you need on exam day.
Start Free Mock TestExam Shortcuts (Pro-Tips)
Matrices and Determinants is a calculation-heavy chapter where examiners reward students who recognise patterns over those who grind through expansions. The seven hacks below collapse 2–3 minutes of work into seconds — every one is grounded in a tested NDA property.
Shortcut 1 — Recognise Special Matrices on Sight
NDA loves vocabulary-based statement questions. Lock these definitions into instant recall:
Involutory: $$A^2 = I$$ | Nilpotent: $$A^k = O$$ for some positive integer k
Symmetric: $$A^T = A$$ | Skew-symmetric: $$A^T = -A$$ (diagonal = 0)
Shortcut 2 — 2×2 Inverse in One Step
Never use the cofactor-then-transpose route for a 2×2 inverse. Use the direct formula:
Swap a and d, flip signs of b and c, divide by the determinant. Done in 10 seconds.
Shortcut 3 — Skew-Symmetric Odd-Order Hack
The determinant of a skew-symmetric matrix of odd order (3×3, 5×5, …) is always zero. If you spot a matrix with all-zero principal diagonal and $$a_{ij} = -a_{ji}$$ off-diagonal, mark 0 without expanding.
Shortcut 4 — |kA| = kⁿ|A| (Not k|A|)
For an n×n matrix, scaling by k pulls out k from each of n rows, giving $$|kA| = k^n |A|$$ — not $$k|A|$$. This is the #1 NDA trap on |3A|-style questions.
Shortcut 5 — Row/Column Operations to Force a Zero Row
If a 3×3 determinant has rows or columns with a visible linear pattern (e.g., each row sum equals the same value, or columns in arithmetic progression), apply $$R_i \to R_i - R_j$$ or $$C_i \to C_i - C_j$$ to create two identical or proportional rows — instantly making $$|A| = 0$$.
If each row sums to the same value s, apply $$C_1 \to C_1 + C_2 + C_3$$ to pull s out as a common factor.
Shortcut 6 — Cramer's Rule Sign Check
For a 2×2 system $$ax + by = e, \; cx + dy = f$$:
Shortcut 7 — Adjoint Determinant Formula
$$|\text{adj}(A)| = |A|^{n-1}$$ is the single most-repeated NDA formula in this chapter. For a 3×3 with |A| = k, |adj A| = k². For |adj(adj A)|, use $$|A|^{(n-1)^2}$$ → for n = 3, that's $$|A|^4$$.
Shortcut 8 — Value-Putting Trick for Algebraic Determinants
If a 3×3 determinant is loaded with variables a, b, c and the options are also algebraic, never expand symbolically. Plug a = 1, b = 2, c = 3 (avoid values that make multiple options equal or denominators zero), compute the numerical determinant, then test the four options with the same values. The match is your answer — usually solved in under 30 seconds.
Shortcut 9 — Cayley-Hamilton for 2×2 Powers
For any 2×2 matrix A, the Cayley-Hamilton identity gives $$A^2 = \text{tr}(A)\cdot A - |A|\cdot I$$. This collapses repeated 2×2 multiplications (like finding $$A^2$$, $$A^3$$) into a single linear combination — a huge time-saver in algebraic matrix-equation problems.
Common Question Patterns
Pattern 1 — Statement-based property questions
Two or three statements about matrices or determinants are given and you must identify which are correct. Common true statements: \(A + A^T\) is symmetric; \(A - A^T\) is skew-symmetric; the inverse of a matrix is unique; a unit matrix is a diagonal matrix. Common false statements: \(\det(A+B) = \det A + \det B\); the product of two non-zero matrices cannot be zero (false — it can be zero if one is singular).
Pattern 2 — Invertibility and adj A
You are told \(|A|\) or the order and asked for \(|\text{adj}\,A|\), \(A^{-1}\), or whether \(A\) is invertible. Always apply \(|\text{adj}\,A| = |A|^{n-1}\). For a 3×3 matrix: \(|\text{adj}\,A| = |A|^2\). If \(A^2 = I\), then \(A^{-1} = A\).
Pattern 3 — Determinant evaluation with properties
You are given a structured 3×3 determinant (often with variables, cube roots of unity, or a pattern like \(a+b+c\) in each row) and asked for its value. The answer is almost always 0 when two rows become identical after a simple row operation, or a factorable expression like \(abc\) or \(a+b+c\).
Pattern 4 — System of equations (Cramer's rule)
Given a 2×2 or 3×3 system, find the value of a parameter for which: (a) the system has no solution, (b) infinitely many solutions, or (c) a unique solution. Always compute \(D\) first. If \(D = 0\), check \(D_1\) and \(D_2\).
Pattern 5 — Order and product definitions
Questions like "for which order is \(AA\) defined?" or "what is the order of \([x\ y\ z]\,[h\ b\ f]^T\,[x\ y\ z]^T\)?" — these test your understanding of when matrix multiplication is valid. The product \(AA\) is defined when \(A\) is \(m \times n\) only if \(m = n\) (i.e., \(A\) is square).
Preparation Strategy
Week 1 — Build the foundation
Read the NCERT Chapter 3 (Matrices) and Chapter 4 (Determinants) definitions and examples carefully. Focus on classifying matrix types and understanding why each property holds, not just memorising it. Do Exercises 3.1 to 3.4 fully. By the end of the week, you should be able to multiply two 3×3 matrices, compute a 3×3 determinant by cofactor expansion, and state at least six determinant properties from memory.
Week 2 — Properties and formulas
Drill the formula cards on this page until they are automatic. Particular priority: \(|\text{adj}\,A| = |A|^{n-1}\), \(A^{-1} = \text{adj}(A)/|A|\), and the \(|kA| = k^n |A|\) rule. These appear in PYQs repeatedly. Practice 10–15 statement-based problems per day from the PYQ file.
Week 3 — PYQ sets and speed
Solve the complete PYQ set for this chapter in timed conditions. Target 90 seconds per question. Review every mistake by categorising it: wrong formula, wrong property, or calculation error. Each category needs a different fix. Revisit the NDA Maths topic list and cross-check which adjacent topics (like Vector Algebra and Three-Dimensional Geometry) also use matrix/determinant ideas — solving those questions reinforces this chapter.
Linked topics to study alongside
Matrices and Determinants connects directly to Sets, Relations and Functions (function composition mirrors matrix multiplication), Quadratic Equations and Inequalities (characteristic equation of a matrix), and Limits, Continuity and Differentiability (L'Hopital-style determinant problems). Covering those pages on NDA Maths in parallel sharpens your overall algebra instincts.
Frequently Asked Questions
How many questions come from Matrices and Determinants in the NDA exam?
Based on PYQ data from 2010 to 2015, each NDA Paper I sitting carries roughly 6–10 questions from this topic across both sets. It is consistently one of the top three algebra topics by question count.
What is the difference between a matrix and a determinant?
A matrix is an ordered rectangular array of numbers — it is a mathematical object, not a number. A determinant is a scalar value (a number) associated with a square matrix. You can only compute a determinant for square matrices; a matrix of any order exists.
When does a matrix not have an inverse?
A square matrix \(A\) is not invertible (singular) when its determinant \(|A| = 0\). For example, if the system of equations formed by \(A\) is inconsistent or has infinitely many solutions, \(|A| = 0\). A non-square matrix is never invertible in the standard sense.
What is the determinant of an odd-order skew-symmetric matrix?
It is always zero. This is a standard property: for a skew-symmetric matrix \(A\) of odd order \(n\), \(|A| = |A^T| = |-A| = (-1)^n |A|\). When \(n\) is odd, this forces \(|A| = 0\). This appeared directly as a 2014 PYQ.
What does |adj A| equal for a 3×3 matrix with |A| = 8?
Use the formula \(|\text{adj}\,A| = |A|^{n-1}\). For \(n = 3\) and \(|A| = 8\): \(|\text{adj}\,A| = 8^2 = 64\). This exact question appeared in the 2010 NDA paper.
How do I decide whether a system has no solution, a unique solution, or infinitely many?
Compute \(D = |A|\) (the coefficient matrix determinant). If \(D \neq 0\), there is a unique solution. If \(D = 0\), compute \(D_1\) and \(D_2\) (replace each column with the constant column in turn). If any \(D_i \neq 0\), the system has no solution. If all \(D_i = 0\), the system either has infinitely many solutions or no solution — check the augmented matrix rank to distinguish.
Is the product of two non-zero matrices always non-zero?
No. If \(A\) and \(B\) are non-zero matrices and \(AB = 0\), it is possible that either \(|A| = 0\) or \(|B| = 0\) (i.e., at least one is singular). This is a classic PYQ trap — the product of two non-zero matrices can be the zero matrix if at least one factor is singular.