Full Chapter Summary & Detailed Notes - Determinants Class 12 NCERT

All mathematical truths are relative and conditional. — C.P. STEINMETZ

4.1 Introduction

In the previous chapter, we have studied about matrices and algebra of matrices. We have also learnt that a system of algebraic equations can be expressed in the form of matrices. This means, a system of linear equations like a1 x + b1 y = c1, a2 x + b2 y = c2 can be represented as \begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}. Now, this system of equations has a unique solution or not, is determined by the number a1 b2 – a2 b1. (Recall that if a1/b1 ≠ a2/b2 or a1 b2 – a2 b1 ≠ 0, then the system of linear equations has a unique solution). The number a1 b2 – a2 b1 which determines uniqueness of solution is associated with the matrix A = \begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix} and is called the determinant of A or det A. Determinants have wide applications in Engineering, Science, Economics, Social Science, etc.

In this chapter, we shall study determinants up to order three only with real entries. Also, we will study various properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle, adjoint and inverse of a square matrix, consistency and inconsistency of system of linear equations and solution of linear equations in two or three variables using inverse of a matrix.

4.2 Determinant

To every square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A, where aij = (i, j)th element of A. This may be thought of as a function which associates each square matrix with a unique number (real or complex). If M is the set of square matrices, K is the set of numbers (real or complex) and f : M → K is defined by f (A) = k, where A ∈ M and k ∈ K, then f (A) is called the determinant of A. It is also denoted by |A| or det A or ∆.

If A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, then determinant of A is written as |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = det (A)

Remarks

• (i) For matrix A, |A| is read as determinant of A and not modulus of A.
• (ii) Only square matrices have determinants.

4.2.1 Determinant of a matrix of order one

Let A = [a] be the matrix of order 1, then determinant of A is defined to be equal to a.

4.2.2 Determinant of a matrix of order two

Let A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} be a matrix of order 2 × 2, then the determinant of A is defined as: det (A) = |A| = ∆ = a11 a22 – a21 a12

4.2.3 Determinant of a matrix of order 3 × 3

Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and C3) giving the same value as shown below.

Consider the determinant of square matrix A = [aij]3 × 3

i.e., |A| = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}

Expansion along first Row (R1)

Step 1: Multiply first element a11 of R1 by (–1)^{1+1} and with the second order determinant obtained by deleting the elements of first row (R1) and first column (C1) of |A| as a11 lies in R1 and C1, i.e., (–1)^{1+1} a11 \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}

Step 2: Multiply 2nd element a12 of R1 by (–1)^{1+2} and the second order determinant obtained by deleting elements of first row (R1) and 2nd column (C2) of |A| as a12 lies in R1 and C2, i.e., (–1)^{1+2} a12 \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}

Step 3: Multiply third element a13 of R1 by (–1)^{1+3} and the second order determinant obtained by deleting elements of first row (R1) and third column (C3) of |A| as a13 lies in R1 and C3, i.e., (–1)^{1+3} a13 \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}

Step 4: Now the expansion of determinant of A, that is, |A| written as sum of all three terms obtained in steps 1, 2 and 3 above is given by det A = |A| = (–1)^{1+1} a11 \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} + (–1)^{1+2} a12 \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + (–1)^{1+3} a13 \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}

or |A| = a11 (a22 a33 – a32 a23) – a12 (a21 a33 – a31 a23) + a13 (a21 a32 – a31 a22) = a11 a22 a33 – a11 a32 a23 – a12 a21 a33 + a12 a31 a23 + a13 a21 a32 – a13 a31 a22 ... (1)

Expansion along second row (R2)

|A| = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}

Expanding along R2, we get |A| = – a21 (a12 a33 – a32 a13) + a22 (a11 a33 – a31 a13) – a23 (a11 a32 – a31 a12)

|A| = – a21 a12 a33 + a21 a32 a13 + a22 a11 a33 – a22 a31 a13 – a23 a11 a32 + a23 a31 a12 = a11 a22 a33 – a11 a23 a32 – a12 a21 a33 + a12 a23 a31 + a13 a21 a32 – a13 a31 a22 ... (2)

Expansion along first Column (C1)

By expanding along C1, we get |A| = a11 (a22 a33 – a23 a32) – a21 (a12 a33 – a13 a32) + a31 (a12 a23 – a13 a22)

|A| = a11 a22 a33 – a11 a23 a32 – a21 a12 a33 + a21 a13 a32 + a31 a12 a23 – a31 a13 a22 = a11 a22 a33 – a11 a23 a32 – a12 a21 a33 + a12 a23 a31 + a13 a21 a32 – a13 a31 a22 ... (3)

Clearly, values of |A| in (1), (2) and (3) are equal. It is left as an exercise to the reader to verify that the values of |A| by expanding along R3, C2 and C3 are equal to the value of |A| obtained in (1), (2) or (3). Hence, expanding a determinant along any row or column gives same value.

Remarks

• (i) For easier calculations, we shall expand the determinant along that row or column which contains maximum number of zeros.
• (ii) While expanding, instead of multiplying by (–1)^{i+j}, we can multiply by +1 or –1 according as (i + j) is even or odd.
• (iii) Let A = \begin{bmatrix} 4 & 2 \\ 0 & 2 \end{bmatrix} and B = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}. Then, it is easy to verify that A = 2B. Also |A| = 8 – 0 = 8 and |B| = 2 – 0 = 2. Observe that, |A| = 4(2) = 2^2 |B| or |A| = 2^n |B|, where n = 2 is the order of square matrices A and B. In general, if A = kB where A and B are square matrices of order n, then |A| = k^n |B|, where n = 1, 2, 3.

4.3 Properties of Determinants

In this section, we shall study some properties of determinants which are very useful in evaluating determinants.

The value of the determinant remains unchanged if its rows and columns are interchanged.

If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.

If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero.

If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.

If to each element of any row or any column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same.

4.4 Area of a Triangle

In earlier classes, we have studied that the area of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3), is given by the expression \frac{1}{2} [x1(y2–y3) + x2 (y3–y1) + x3 (y1–y2)]. Now this expression can be written in the form of a determinant as ∆ = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} ... (1)

Remarks

• (i) Since area is a positive quantity, we always take the absolute value of the determinant in (1).
• (ii) If area is given, use both positive and negative values of the determinant for calculation.
• (iii) The area of the triangle formed by three collinear points is zero.

4.5 Minors and Cofactors

In this section, we will learn to write the expansion of a determinant in compact form using minors and cofactors.

Definition 1 Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column in which element aij lies. Minor of an element aij is denoted by Mij.

Remark: Minor of an element of a determinant of order n(n ≥ 2) is a determinant of order n – 1.

Definition 2 Cofactor of an element aij, denoted by Aij is defined by Aij = (–1)^{i+j} Mij, where Mij is minor of aij.

4.6 Adjoint and Inverse of a Matrix

In the previous chapter, we have studied inverse of a matrix. In this section, we shall discuss the condition for the existence of inverse of a matrix.

To find inverse of a matrix A, i.e., A^{-1} we shall first define adjoint of a matrix.

Definition 3 The adjoint of a square matrix A = [aij]_{n×n} is defined as the transpose of the matrix [Aij]_{n×n}, where Aij is the cofactor of the element aij. Adjoint of the matrix A is denoted by adj A.

Let A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}. The cofactor matrix of A is \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix}. The adjoint of A is the transpose of the cofactor matrix, i.e., adj A = \begin{bmatrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end{bmatrix}.

Theorem 1 If A be any given square matrix of order n, then A (adj A) = (adj A) A = |A| I, where I is the identity matrix of order n.

Theorem 2 A square matrix A is invertible if and only if |A| ≠ 0.

A^{-1} = \frac{1}{|A|} adj A

4.7 Applications of Determinants and Matrices

In this section, we shall discuss application of determinants and matrices for solving the system of linear equations in two or three variables and for checking the consistency of the system of linear equations.

Consistent system A system of linear equations is said to be consistent, if it has at least one solution.

Inconsistent system A system of linear equations is said to be inconsistent, if it has no solution.

Consider the system of equations a1 x + b1 y + c1 z = d1, a2 x + b2 y + c2 z = d2, a3 x + b3 y + c3 z = d3.

Let A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}, X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix}.

Then, the system can be written as A X = B.

Case I If |A| ≠ 0, then system is consistent and has a unique solution given by X = A^{-1} B.

Case II If |A| = 0 and (Adj A) B ≠ O, then system is inconsistent.

Case III If |A| = 0 and (Adj A) B = O, then system may be consistent with infinite solutions or inconsistent (i.e., no solution).

Summary & Exercises Tease

Key Takeaways: Det measures volume/scaling; properties simplify eval; adjoint for inverse; systems consistency via det/adj. Exercises: Basics (4.1), Area (4.2), Properties (4.3), Minors/Adj (4.4), Systems (4.5), Misc advanced.

Key Definitions & Terms - Complete Glossary (Expanded 25+ Terms)

All terms from NCERT Chapter 4, detailed with examples, relevance, proofs where applicable. Grouped by subtopic for easy flashcards. Added advanced terms like "Singular Matrix". Historical notes (Laplace expansion). Real-life: Det in computer graphics for transformations. Use MathJax for book-like rendering.

Tip: Flashcards: Term front, Ex + Relevance back. Depth: Prove Property 3 from expansion. Interlinks: Det to inverse from Ch3. Graphs: Visualize det as signed area.

Solved Examples from NCERT - Step-by-Step with MathJax (Full Book Examples)

All 15+ NCERT examples solved in detail, grouped by section. Expanded with steps, diagrams desc, verification. Mirrors book layout with equations rendered.

Tip: Verify det with multiple expansions; systems check det first. Practice 3x3 full.

Exercise 4.1 Questions & Answers - Full 8 Qs with MathJax (Book Exact)

Full solutions for all 8 questions on basics/2x2/3x3 dets. Step-by-step like solved examples. Expanded explanations.

Tip: For 3x3, choose row/col with zeros; verify with another expansion.

Exercise 4.2 Questions & Answers - Full 5 Qs on Area/Collinear

Area formula, collinear checks. Full with coords plug-in.

Tip: Abs for area; det=0 collinear. For eq, set det=0 with (x,y,1).

Exercise 4.3 Questions & Answers - Full 5 Qs on Properties

Prove/use properties to eval dets. Full proofs.

Tip: Props reduce computation; look for factors, zeros.

Exercise 4.4 Questions & Answers - Full 18 Qs on Minors/Cof/Adj/Inv

Minors, cofactors, adj, inverse calcs. Full 3x3.

Tip: Sign checkerboard; verify A A^{-1}=I. Ops for inv efficient.

Exercise 4.5 Questions & Answers - Full 14 Qs on Systems

Consistency, solve using inverse/Cramer. Full systems.

Tip: det A first; if 0 check adj B. Cramer for small n.

Miscellaneous Exercise Questions & Answers - Full 18 Qs Advanced

Mix: Props proofs, eval, systems, apps. Full proofs/calcs.

Tip: Misc tests depth; proofs 4-6 marks. Apps real-world.

Quick Revision Notes & Mnemonics (Table Expanded)

Concise 1-page scan; bold keys, phrases. Full chapter in 5 rows. Updated with properties proofs.

Overall Mnemonic: BPACM-IS (scan 3 mins). Flashcards: 1 per row. 2025 Exam: 3x3 eval (4 marks), system (6 marks), props proofs (2 marks).