Class 12 Maths NCERT Solutions Chapter 4 Determinants | Boundless Maths

Unit II · Algebra · Chapter 4 Determinants — Class 12 Maths NCERT Solutions

Class 12 Maths NCERT Solutions Chapter 4: Determinants

Free, step-by-step NCERT Solutions for all five exercises of this chapter, covering everything from evaluating a simple 2×2 determinant to solving a three-variable word problem by the matrix method. You'll use the same handful of ideas — expansion along a row or column, minors and cofactors, the adjoint — to find the area of a triangle from its vertices, prove three points are collinear, compute the inverse of a matrix, and finally decide whether a system of linear equations has a unique solution at all. Every solution here is worked the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year laid out right on this page.

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6Exercises (incl. Misc.)

61Total Questions

2026-27CBSE Syllabus

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Class 12 Maths NCERT Solutions Chapter 4 — Overview

This Class 12 Maths NCERT Solutions Chapter 4 hub covers Determinants, the natural sequel to Chapter 3 that assigns a single real number to every square matrix. That number — computed by expanding along any row or column — tells you at a glance whether a system of linear equations has a unique solution, and it's the engine behind the area of a triangle, the adjoint of a matrix, and ultimately the inverse of a matrix. Where Chapter 3 taught you to manipulate matrices, this chapter teaches you to extract a decisive piece of information from one.

The chapter builds in a deliberate sequence: evaluating determinants of order 1, 2 and 3 first (Exercise 4.1), then a geometric application to the area of a triangle and testing collinearity (Exercise 4.2), then minors and cofactors as the building blocks of expansion (Exercise 4.3), then the adjoint and inverse of a matrix built from those cofactors (Exercise 4.4), and finally the payoff — solving a system of two or three linear equations by the matrix method, and checking whether that system is consistent (Exercise 4.5). Getting comfortable with cofactor expansion here pays off directly when you reach vectors and three-dimensional geometry later in the syllabus.

How the Chapter Builds

One Idea Leads to the Next

Determinants (Order 1–3)

A single number associated with a square matrix, evaluated by expansion along a row or column. Exercise 4.1.

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Area of a Triangle

A determinant built from three vertices gives the area directly — and a zero area means the points are collinear. Exercise 4.2.

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Minors, Cofactors & Adjoint

Minors and signed cofactors combine into the adjoint matrix — the key ingredient for the inverse. Exercise 4.3 & 4.4.

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Solving Linear Systems

Write the system as AX = B and solve X = A⁻¹B — provided A is nonsingular. Exercise 4.5.

Quick Reference

Important Formulas — Chapter 4

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Algebra.

Determinants of Order 1, 2 and 3 (§4.2)

Order 1 and order 2

|a| = a,\qquad \begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix} = a_{11}a_{22} - a_{21}a_{12}

The order-2 rule is the cross-multiply-and-subtract shortcut you'll use constantly.

Order 3 (expansion along R₁)

|A| = a_{11}(a_{22}a_{33}-a_{32}a_{23}) - a_{12}(a_{21}a_{33}-a_{31}a_{23}) + a_{13}(a_{21}a_{32}-a_{31}a_{22})

Expanding along any row or column gives the same value — always pick the one with the most zeros.

Scalar multiple of a matrix

|kA| = k^{n}\,|A|

n is the order of the square matrix — a very common trap when only one row is scaled by k instead of the whole matrix.

Area of a Triangle (§4.3)

Area from three vertices

\Delta = \tfrac{1}{2}\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}

Always take the absolute value — area can never be negative, even if the determinant is.

Collinearity test

\Delta = 0 \iff \text{the three points are collinear}

A zero-area "triangle" is the standard way this chapter tests three points lying on one line.

Minors, Cofactors, Adjoint & Inverse (§4.4–4.5)

Minor & cofactor

A_{ij} = (-1)^{i+j}\,M_{ij}

M_ij is the determinant left after deleting row i and column j; the cofactor just adds the alternating sign.

Expansion via cofactors

|A| = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}

Elements of a row times cofactors of a different row always sum to zero.

Adjoint & inverse

A(\text{adj}\,A) = (\text{adj}\,A)A = |A|\,I,\qquad A^{-1} = \dfrac{1}{|A|}\,\text{adj}\,A

adj A is the transpose of the cofactor matrix — the inverse exists only when |A| ≠ 0.

Solving Systems of Linear Equations (§4.6)

Matrix method

AX = B,\quad |A| \neq 0 \;\Rightarrow\; X = A^{-1}B

Gives the unique solution directly — this is the case tested throughout Exercise 4.5.

Checking consistency when |A| = 0

(adj A)B ≠ O → no solution (inconsistent). (adj A)B = O → infinitely many solutions or no solution.

NCERT restricts full solving to the unique-solution case, but this test still shows up in consistency questions.

Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all five exercises.

Confusing |A| with A. |A| is read "determinant of A" — a single number — never "modulus of A," and only a square matrix has a determinant at all.
Sign errors in cofactor expansion. The alternating sign (−1)^i+j is easy to drop or misapply, especially past the first row — write out the sign pattern (+,−,+ / −,+,− / +,−,+) before you start.
Misapplying the scalar-multiple rule. |kA| = kⁿ|A| applies only when every entry of the matrix is scaled by k. If just one row is multiplied by k, the determinant is only k|A|, not kⁿ|A|.
Forgetting to take the absolute value in the area-of-a-triangle formula. The determinant can come out negative depending on vertex order — area itself is always positive.
Mixing up a minor with a cofactor. M_ij is just the smaller determinant; A_ij requires the extra (−1)^i+j sign — stating one when the question asks for the other is a common slip.
Using cofactors from the wrong row or column when expanding. Elements of row i times cofactors of row i give |A|; elements of row i times cofactors of a different row always give zero — don't mix the two.
Assuming every square matrix is invertible. A^-1 exists only when A is nonsingular, i.e. |A| ≠ 0 — always evaluate |A| before attempting adj A / |A|.
Applying the matrix method when |A| = 0. This chapter's matrix method solves only systems with a unique solution. When |A| = 0, you must instead test (adj A)B to decide inconsistency versus infinitely many solutions.

Solve Chapter-Wise

Class 12 Maths NCERT Solutions Chapter 4 — Choose an Exercise

4.1

Exercise 4.1

Evaluating determinants of order 1, 2 and 3, and using scalar-multiple properties · 8 questions

Solve Exercise 4.1 →

4.2

Exercise 4.2

Area of a triangle from its vertices, proving collinearity, and finding an unknown coordinate · 5 questions

Solve Exercise 4.2 →

4.3

Exercise 4.3

Writing minors and cofactors, and evaluating determinants using cofactor expansion · 5 questions

Solve Exercise 4.3 →

4.4

Exercise 4.4

Finding the adjoint, verifying A(adj A) = |A|I, and computing the inverse of a matrix · 18 questions

Solve Exercise 4.4 →

4.5

Exercise 4.5

Checking consistency and solving systems of linear equations by the matrix method · 16 questions

Solve Exercise 4.5 →

Miscellaneous Exercise

Mixed questions combining determinant properties, adjoint identities, and matrix-method word problems · 9 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Algebra — matrices, determinants — in one printable PDF.

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Common Questions

Frequently Asked Questions

Quick answers about Chapter 4, Determinants.

How many exercises are there in Chapter 4, Determinants?

There are five main exercises — 4.1, 4.2, 4.3, 4.4 and 4.5 — plus a Miscellaneous Exercise, totalling 61 questions across determinant evaluation, area of a triangle, minors and cofactors, adjoint and inverse, and solving systems of linear equations.

Is this chapter important for the board exam?

Yes — Determinants builds directly on Matrices to complete the Algebra unit, and questions on evaluating determinants, finding the inverse via adjoint, and solving a 3-variable system by the matrix method appear almost every year, often as a 5-mark question.

What is the difference between a minor and a cofactor?

The minor M_ij of an element a_ij is the determinant left after deleting that element's row and column. The cofactor A_ij is the same minor with a sign attached: A_ij = (−1)^i+j M_ij. The sign is the part students most often forget.

When does a system of linear equations have a unique solution using the matrix method?

Writing the system as AX = B, a unique solution X = A⁻¹B exists exactly when A is nonsingular, that is, |A| ≠ 0. If |A| = 0, the matrix method as taught in this chapter doesn't apply directly — you instead check (adj A)B to decide whether the system is inconsistent or has infinitely many solutions.

What should I revise before starting this chapter?

Chapter 3, Matrices, is essential preparation — Determinants leans directly on matrix multiplication, transpose, and the definition of an invertible matrix, and several exercises ask you to verify identities like A(adj A) = |A| I using matrix multiplication from the previous chapter.

Where can I find the official NCERT textbook for this chapter?

Determinants is Chapter 4 of the NCERT Class 12 Mathematics textbook (Part I), published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.