When expanding a determinant, must I always use the first row?

Chapter 5 of 18

28%

CBSE Class 12 Mathematics★ 10-13 marks · CBSE Class 12 Mathematics Chapter 4 (combined with Matrices unit)

Determinants

A DETERMINANT is a single number computed from a square matrix — and it tells you whether the matrix is invertible, whether a system of equations has a unique solution, and much more. CBSE Class 12 Mathematics Chapter 4.

⏱ 20 min readHard difficulty✓ Free · NCERT-aligned

Ask AI about this

🔤Build your vocabulary from this chapterLexiGenome mines the key words, explains them in your language, and schedules them with spaced repetition.Mine words →

By the end of this chapter you'll be able to…

1Evaluate determinants of 2×2 and 3×3 matrices using the expansion method and properties
2Apply determinant properties to simplify computation: row/column operations preserve value
3Find the area of a triangle using the determinant formula
4Find the adjoint and inverse of a matrix using cofactors: A⁻¹ = adj(A) / |A|
5Use Cramer's Rule / matrix method to solve a system of linear equations

💡

Why this chapter matters

Determinants is the high-computation chapter of Class 12 Mathematics. Area of a triangle, consistency of systems, and finding inverses via adjoint are all board-exam guaranteed. The Cramer's Rule connection makes determinants directly applicable to linear systems. Cofactors and adjoint must be computed without error.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

🔗

Determinants

"If the determinant is zero, the matrix is singular — no inverse exists. The system is either inconsistent or has infinite solutions."

1. Chapter Overview

A DETERMINANT is a SCALAR value that can be computed from a square matrix. This chapter covers: computing determinants (expansion, properties), the RELATIONSHIP between determinants and matrices, CRAMER'S RULE for solving systems of linear equations, and finding the INVERSE of a matrix using the adjoint method (A⁻¹ = adj(A)/|A|, provided |A| ≠ 0).

2. Computing Determinants

For a 3×3 Matrix (Expansion)

|A| = a₁₁(cofactor of a₁₁) + a₁₂(cofactor of a₁₂) + a₁₃(cofactor of a₁₃)

Cofactor

Minor Mᵢⱼ = determinant of submatrix after removing row i, column j
Cofactor Aᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ

3. Properties of Determinants

Property	What It Means
**	A
If two rows/columns are interchanged	Sign CHANGES
If two rows/columns are identical
If each element of a row is multiplied by k	Determinant is multiplied by k
If a row is added to a multiple of another row	Determinant UNCHANGED

4. Area of a Triangle Using Determinants

Area = ½ |det of the 3×3 matrix formed by vertices|. If area = 0 → points are COLLINEAR.

5. Adjoint and Inverse

Adjoint of A = Transpose of the cofactor matrix
A⁻¹ = adj(A) / |A| — provided |A| ≠ 0
A × adj(A) = adj(A) × A = |A| × I

6. Solving Linear Equations — Cramer's Rule / Matrix Method

Write system as AX = B. If |A| ≠ 0 → unique solution: X = A⁻¹B.
Cramer's Rule: xᵢ = |Aᵢ|/|A|, where Aᵢ replaces column i of A with B.

7. Exam Focus

Determinant expansion. Properties.
Adjoint and inverse: A⁻¹ = adj(A)/|A|. Condition: |A| ≠ 0.
Solving AX = B. Consistent if |A| ≠ 0 (unique solution).
Area of triangle using determinants.

8. Conclusion

Determinants are the GATEWAY to solving systems:

|A| ≠ 0: Matrix is INVERTIBLE. System has a UNIQUE solution.
|A| = 0: Matrix is SINGULAR. No inverse. System has either NO solution or INFINITE solutions.
'The determinant is the single most important number associated with a square matrix.'

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Determinant Expansion

2×2: |A| = |a b; c d| = ad − bc. 3×3 (expanding along first row): |A| = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁). SIGN PATTERN: + − + / − + − / + − + (checkerboard). COFACTOR Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ where Mᵢⱼ is the MINOR (determinant of submatrix with row i, col j removed).

Can expand along ANY row or column — choose the one with the most zeros to minimise computation. 'Expand along the row/column with most zeros.'

Key Determinant Properties

1. Interchanging two rows/columns: |A| changes sign. 2. Row/column of zeros: |A| = 0. 3. Two identical rows/columns: |A| = 0. 4. Scalar multiplication of ONE row/column by k: |A| multiplied by k. (NOT the whole matrix.) 5. |kA| = kⁿ|A| for n×n matrix. 6. Row/column addition: Adding k times one row to another → |A| unchanged. 7. |AB| = |A||B|. 8. |Aᵀ| = |A|. 9. |A⁻¹| = 1/|A|.

The most useful property for computation: adding a multiple of one row to another doesn't change the determinant — use this to create zeros, then expand along the zero-heavy row/column.

Adjoint, Inverse, and Area Formula

COFACTOR MATRIX: Cᵢⱼ = (−1)^(i+j) Mᵢⱼ. ADJOINT: adj(A) = (cofactor matrix)ᵀ (TRANSPOSE of cofactor matrix). INVERSE: A⁻¹ = adj(A) / |A|. Valid only when |A| ≠ 0. AREA OF TRIANGLE: Given vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): Area = ½|det([[x₁,y₁,1],[x₂,y₂,1],[x₃,y₃,1]])|. COLLINEAR TEST: Three points are collinear ⟺ the above determinant = 0.

adj(A) = TRANSPOSE of cofactor matrix. The most common error is forgetting to TRANSPOSE — computing the cofactor matrix but not transposing it. For 2×2: adj([[a,b],[c,d]]) = [[d,−b],[−c,a]] (swap diagonal, negate off-diagonal).

System of Equations — Matrix Method

AX = B where A is coefficient matrix, X is variable vector, B is constants vector. SOLUTION: X = A⁻¹B (if |A| ≠ 0). CONSISTENT (unique solution): |A| ≠ 0. INCONSISTENT (no solution): |A| = 0 and (adj A)B ≠ 0. INFINITELY MANY SOLUTIONS: |A| = 0 and (adj A)B = 0.

Steps for 3-equation systems: (1) Write A, X, B. (2) Calculate |A|. (3) If |A| ≠ 0: find A⁻¹ via cofactors. (4) X = A⁻¹B. Always verify your solution by substituting back.

⚠️

Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Forgetting to transpose the cofactor matrix to get adj(A)

✓ adj(A) = TRANSPOSE of the cofactor matrix. STEP 1: find all 9 cofactors Cᵢⱼ. STEP 2: arrange them in a 3×3 cofactor matrix. STEP 3: TRANSPOSE it (row i, col j of cofactor matrix becomes row j, col i of adj(A)). Many students correctly compute cofactors but write them in the wrong position because they forget the transpose.

WATCH OUT

✗ Applying |kA| = k|A| instead of |kA| = kⁿ|A|

✓ |kA| = kⁿ|A| where n is the ORDER of the matrix. For a 3×3 matrix: |kA| = k³|A|. Each of the 3 rows is multiplied by k, and each row multiplication multiplies the determinant by k. So total multiplication = k × k × k = k³. |kA| = k²|A| for 2×2. Never write |kA| = k|A| (which would be true for 1×1 only).

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex s 4.1–4.6

Exercises 4.1–4.6

Determinant evaluation; properties proofs; area of triangle; adjoint and inverse; solving systems of 3 equations in 3 unknowns

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· determinant-properties

Using properties of determinants, prove: |a+b, a, a; a, a+b, a; a, a, a+b| = b²(3a+b).

▶ Show solution

Apply C₁ → C₁ + C₂ + C₃: First column becomes: a+b+a+a = 3a+b (same for all rows). = (3a+b)|1, a, a; 1, a+b, a; 1, a, a+b|. Factor (3a+b) out of C₁. R₂ → R₂ − R₁, R₃ → R₃ − R₁: = (3a+b)|1, a, a; 0, b, 0; 0, 0, b|. Expand along C₁: = (3a+b) × 1 × |b, 0; 0, b| = (3a+b) × b² = b²(3a+b). ✓

Q2MEDIUM· inverse-adjoint

Find the inverse of A = [[1, 2, 1], [3, 1, −1], [2, 0, 1]] using cofactors.

▶ Show solution

|A| = 1(1×1−(−1)×0) − 2(3×1−(−1)×2) + 1(3×0−1×2) = 1(1) − 2(3+2) + 1(0−2) = 1 − 10 − 2 = −11. Since |A| = −11 ≠ 0, A⁻¹ exists. COFACTORS: C₁₁ = (+1)|1,−1;0,1| = 1. C₁₂ = (−1)|3,−1;2,1| = −(3+2) = −5. C₁₃ = (+1)|3,1;2,0| = 0−2 = −2. C₂₁ = (−1)|2,1;0,1| = −(2−0) = −2. C₂₂ = (+1)|1,1;2,1| = 1−2 = −1. C₂₃ = (−1)|1,2;2,0| = −(0−4) = 4. C₃₁ = (+1)|2,1;1,−1| = −2−1 = −3. C₃₂ = (−1)|1,1;3,−1| = −(−1−3) = 4. C₃₃ = (+1)|1,2;3,1| = 1−6 = −5. Cofactor matrix = [[1,−5,−2],[−2,−1,4],[−3,4,−5]]. adj(A) = (Cofactor matrix)ᵀ = [[1,−2,−3],[−5,−1,4],[−2,4,−5]]. A⁻¹ = adj(A)/|A| = (1/−11)[[1,−2,−3],[−5,−1,4],[−2,4,−5]].

Q3HARD· long-answer

Using the matrix method, solve the system: x + 2y + z = 7; x + 3z = 11; 2x − 3y = 1.

▶ Show solution

Write as AX = B: A = [[1,2,1],[1,0,3],[2,−3,0]], X = [[x],[y],[z]], B = [[7],[11],[1]]. FIND |A|: Expand along R₁: |A| = 1|0,3;−3,0| − 2|1,3;2,0| + 1|1,0;2,−3| = 1(0+9) − 2(0−6) + 1(−3−0) = 9 + 12 − 3 = 18. |A| = 18 ≠ 0, so unique solution exists. COFACTORS: C₁₁ = |0,3;−3,0| = 9. C₁₂ = −|1,3;2,0| = −(0−6) = 6. C₁₃ = |1,0;2,−3| = −3. C₂₁ = −|2,1;−3,0| = −(0+3) = −3. C₂₂ = |1,1;2,0| = 0−2 = −2. C₂₃ = −|1,2;2,−3| = −(−3−4) = 7. C₃₁ = |2,1;0,3| = 6. C₃₂ = −|1,1;1,3| = −(3−1) = −2. C₃₃ = |1,2;1,0| = 0−2 = −2. adj(A) = (cofactors)ᵀ = [[9,−3,6],[6,−2,−2],[−3,7,−2]]. A⁻¹ = (1/18)[[9,−3,6],[6,−2,−2],[−3,7,−2]]. X = A⁻¹B = (1/18)[[9×7+(−3)×11+6×1],[6×7+(−2)×11+(−2)×1],[−3×7+7×11+(−2)×1]] = (1/18)[[63−33+6],[42−22−2],[−21+77−2]] = (1/18)[[36],[18],[54]] = [[2],[1],[3]]. SOLUTION: x = 2, y = 1, z = 3. VERIFY: 2 + 2 + 3 = 7 ✓; 2 + 9 = 11 ✓; 4 − 3 = 1 ✓.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•2×2 determinant: ad − bc. 3×3: expansion along first row using cofactors.
•Sign pattern for cofactors: (+−+/−+−/+−+). Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ.
•Property: same two rows/cols → |A| = 0. Swapping rows/cols → sign changes.
•|kA| = kⁿ|A| for n×n. |AB| = |A||B|.
•adj(A) = TRANSPOSE of cofactor matrix. A⁻¹ = adj(A)/|A|.
•For 2×2: adj([[a,b],[c,d]]) = [[d,−b],[−c,a]].
•AX = B: |A|≠0 → unique solution X = A⁻¹B. |A|=0 → no solution or infinite solutions.
•Area of triangle = ½|det([[x₁,y₁,1],[x₂,y₂,1],[x₃,y₃,1]])|. Zero = collinear.

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 10-13 marks

Question type	Marks each	Typical count	What it tests
Determinant Proof (properties)	5	1	Prove value of a determinant using row/column operations to simplify
Inverse or System Solution	6	1	Find inverse using cofactors/adjoint; solve 3×3 system using matrix method; consistency check

Prep strategy

For determinant proofs: ALWAYS start by identifying the operation (row/column sum, cyclic, etc.). The key technique is using row operations (Rᵢ → Rᵢ + kRⱼ) to create zeros or factor out, then expand along a zero-heavy row. Show each step.
adj(A) = TRANSPOSE of cofactor matrix — write this reminder at the top of your answer before starting. The most common error is writing cofactors in the wrong positions.
Matrix method for systems: show AX = B setup, calculate |A|, state whether unique solution exists, find A⁻¹, compute X = A⁻¹B, VERIFY. Verification earns the final mark and catches errors.

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Solving Kirchhoff's Circuit Equations

Kirchhoff's Current and Voltage Laws in electrical circuits generate systems of 3-5 linear equations. Engineers solve these using exactly the matrix method in this chapter: write the coefficient matrix, compute the determinant (non-zero = unique current distribution), then solve AX = B for the currents. Every circuit simulation tool (SPICE, LTspice) implements determinant-based linear algebra at its core.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

For determinant proofs: state each row/column operation on a new line (e.g., 'C₁ → C₁ + C₂ + C₃'). Take out common factors immediately after each operation — this keeps numbers small and reduces errors.
For solving systems: ALWAYS verify your solution by substituting x, y, z back into the original equations. This earns 1 mark and guards against arithmetic errors in A⁻¹ calculation.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Study the CHARACTERISTIC POLYNOMIAL of a matrix: det(A − λI) = 0 gives the eigenvalues. The CAYLEY-HAMILTON THEOREM states that every square matrix satisfies its own characteristic polynomial. This is the bridge between determinants and eigenvalue theory — the foundation of advanced linear algebra
Research HOW GOOGLE'S PAGERANK works: it uses the dominant eigenvector of a very large matrix (the web's link structure). The determinant properties and matrix inverse studied in this chapter underpin the algorithm that ranks all web pages

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 12 Board (Mathematics)High

JEE Main (Determinants)High

CUET (Mathematics)High

AI helpers for this chapter

Generate more practice on the fly, or have the chapter explained at a different level.

⚡

Generate 5 more questions

Fresh MCQs + short-answer items based on this chapter, plus answer keys.

🧠

Explain at a different level

Re-explain the chapter at the level you actually want.

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

NO — you can expand along ANY row or column. The result is the same regardless of which row/column you choose (this is a theorem). The STRATEGY is to choose the row/column with the MOST ZEROS, because terms with zero cofactor vanish, reducing computation. Before expanding, use row/column operations to create as many zeros as possible in one row/column.

Practice more, ask doubts, find a tutor

CBSE Class 12 mock tests

CBSE Class 12 PYQs

Ask in Q&A

Mathematics flashcards

Related chapters

Students who read Determinants usually read these next.

Relations and Functions

A RELATION links elements of two sets. A FUNCTION is a special relation where EVERY input has EXACTLY ONE output. This chapter builds the foundation — types of relations, types of functions, composition of functions and invertible functions, and binary operations. CBSE Class 12 Mathematics Chapter 1.

Relations and Functions (Class 12)

A deeper dive into RELATIONS (reflexive, symmetric, transitive — equivalence relations) and FUNCTIONS (one-one, onto, bijective — composition, invertibility). CBSE Class 12 Mathematics Chapter 1.

Inverse Trigonometric Functions

If sin(x) = y, what is x in terms of y? The answer is sin⁻¹(y) — but with a TWIST: since trig functions are NOT one-one, we must RESTRICT their domains to define inverses. CBSE Class 12 Mathematics Chapter 2.

Matrices

A MATRIX is a rectangular array of numbers — and it is one of the most POWERFUL tools in mathematics. This chapter covers types of matrices, operations (addition, multiplication, transpose), symmetric and skew-symmetric matrices, and finding an inverse using elementary operations. CBSE Class 12 Mathematics Chapter 3.