Full Chapter Summary & Detailed Notes - Matrices Class 12 NCERT

The essence of Mathematics lies in its freedom. — CANTOR

3.1 Introduction

The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematical tool simplifies our work to a great extent when compared with other straightforward methods. The evolution of the concept of matrices is the result of an attempt to obtain compact and simple methods of solving systems of linear equations. Matrices are not only used as a representation of the coefficients in a system of linear equations, but the utility of matrices far exceeds that use. Matrix notation and operations are used in electronic spreadsheet programs for personal computers, which in turn are used in different areas of business and science like budgeting, sales projection, cost estimation, analyzing the results of an experiment, etc. Also, many physical operations such as magnification, rotation, and reflection through a plane can be represented mathematically by matrices. Matrices are also used in cryptography. This mathematical tool is not only used in certain branches of sciences, but also in genetics, economics, sociology, modern psychology, and industrial management.

In this chapter, we shall find it interesting to become acquainted with the fundamentals of matrix and matrix algebra.

3.2 Matrix

Suppose we wish to express the information that Radha has 15 notebooks. We may express it as [15] with the understanding that the number inside [ ] is the number of notebooks that Radha has. Now, if we have to express that Radha has 15 notebooks and 6 pens, we may express it as [15 6] with the understanding that the first number inside [ ] is the number of notebooks while the other one is the number of pens possessed by Radha.

Let us now suppose that we wish to express the information of possession of notebooks and pens by Radha and her two friends Fauzia and Simran which is as follows:

• Radha has 15 notebooks and 6 pens,
• Fauzia has 10 notebooks and 2 pens,
• Simran has 13 notebooks and 5 pens.

Now this could be arranged in the tabular form as follows:

\[ \begin{bmatrix} 15 & 6 \\ 10 & 2 \\ 13 & 5 \end{bmatrix} \]

or

\[ \begin{array}{ccc} \text{Radha} & \text{Fauzia} & \text{Simran} \\ 15 & 10 & 13 \\ 6 & 2 & 5 \end{array} \]

In the first arrangement the entries in the first column represent the number of notebooks possessed by Radha, Fauzia and Simran, respectively and the entries in the second column represent the number of pens possessed by Radha, Fauzia and Simran, respectively. Similarly, in the second arrangement, the entries in the first row represent the number of notebooks possessed by Radha, Fauzia and Simran, respectively. The entries in the second row represent the number of pens possessed by Radha, Fauzia and Simran, respectively. An arrangement or display of the above kind is called a matrix. Formally, we define matrix as:

Definition 1 A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

We denote matrices by capital letters. The following are some examples of matrices:

\[ A = \begin{bmatrix} 5 & -2 \\ 0 & 5 \\ 3 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 & 3 \\ 3.5 & -1 & 2 \\ i + 5 & 3 & 5 \end{bmatrix}, \quad C = \begin{bmatrix} 3 + \cos x & \tan x & \sin x \\ 1 & x & x^2 + x^3 \end{bmatrix} \]

In the above examples, the horizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix. Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2 rows and 3 columns.

3.2.1 Order of a matrix

A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix). So referring to the above examples of matrices, we have A as 3 × 2 matrix, B as 3 × 3 matrix and C as 2 × 3 matrix. We observe that A has 3 × 2 = 6 elements, B and C have 9 and 6 elements, respectively.

In general, an m × n matrix has the following rectangular array:

\[ A = [a_{ij}]_{m \times n}, \quad 1 \leq i \leq m, \ 1 \leq j \leq n \quad i, j \in \mathbb{N} \]

Thus the ith row consists of the elements \( a_{i1}, a_{i2}, a_{i3}, \dots, a_{in} \), while the jth column consists of the elements \( a_{1j}, a_{2j}, a_{3j}, \dots, a_{mj} \).

In general \( a_{ij} \), is an element lying in the ith row and jth column. We can also call it as the (i, j)th element of A. The number of elements in an m × n matrix will be equal to mn.

Note In this chapter

• We shall follow the notation, namely \( A = [a_{ij}]_{m \times n} \) to indicate that A is a matrix of order m × n.
• We shall consider only those matrices whose elements are real numbers or functions taking real values.

We can also represent any point (x, y) in a plane by a matrix (column or row) as

\[ \begin{bmatrix} x \\ y \end{bmatrix} \quad (\text{or} \ [x, y]). \]

For example point P(0, 1) as a matrix representation may be given as

\[ P = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad \text{or} \ [0 \ 1]. \]

Observe that in this way we can also express the vertices of a closed rectilinear figure in the form of a matrix. For example, consider a quadrilateral ABCD with vertices A (1, 0), B (3, 2), C (1, 3), D (–1, 2).

Now, quadrilateral ABCD in the matrix form, can be represented as

\[ X = \begin{bmatrix} 1 & 3 & 1 & -1 \\ 0 & 2 & 3 & 2 \end{bmatrix} \quad \text{or} \quad Y = \begin{bmatrix} 1 & 0 \\ 3 & 2 \\ 1 & 3 \\ -1 & 2 \end{bmatrix}. \]

Thus, matrices can be used as representation of vertices of geometrical figures in a plane.

Example 1 (Integrated in Summary - Workers Data)

Consider the following information regarding the number of men and women workers in three factories I, II and III:

\[ \begin{array}{c|cc} & \text{Men workers} & \text{Women workers} \\ \hline \text{I} & 30 & 25 \\ \text{II} & 25 & 31 \\ \text{III} & 27 & 26 \end{array} \]

Represent the above information in the form of a 3 × 2 matrix. What does the entry in the third row and second column represent?

Solution: The information is represented in the form of a 3 × 2 matrix as follows:

\[ A = \begin{bmatrix} 30 & 25 \\ 25 & 31 \\ 27 & 26 \end{bmatrix} \]

The entry in the third row and second column represents the number of women workers in factory III.

Example 2 (Integrated - Possible Orders)

If a matrix has 8 elements, what are the possible orders it can have?

Solution: We know that if a matrix is of order m × n, it has mn elements. Thus, to find all possible orders of a matrix with 8 elements, we will find all ordered pairs of natural numbers, whose product is 8. Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4). Hence, possible orders are 1 × 8, 8 ×1, 4 × 2, 2 × 4.

Example 3 (Integrated - Construct Matrix)

Construct a 3 × 2 matrix whose elements are given by \( a_{ij} = \frac{1}{2} |i - 3j| \).

Solution: In general a 3 × 2 matrix is given by \[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix}. \] Now \( a_{ij} = \frac{1}{2} |i - 3j| \), i = 1, 2, 3 and j = 1, 2. Therefore \( a_{11} = \frac{1}{2} |1 - 3 \cdot 1| = \frac{1}{2} \cdot 2 = 1 \), \( a_{12} = \frac{1}{2} |1 - 3 \cdot 2| = \frac{1}{2} \cdot 5 = \frac{5}{2} \), \( a_{21} = \frac{1}{2} |2 - 3 \cdot 1| = \frac{1}{2} \cdot 1 = \frac{1}{2} \), \( a_{22} = \frac{1}{2} |2 - 3 \cdot 2| = \frac{1}{2} \cdot 4 = 2 \), \( a_{31} = \frac{1}{2} |3 - 3 \cdot 1| = \frac{1}{2} \cdot 0 = 0 \), \( a_{32} = \frac{1}{2} |3 - 3 \cdot 2| = \frac{1}{2} \cdot 3 = \frac{3}{2} \). Hence the required matrix is given by \[ A = \begin{bmatrix} 1 & \frac{5}{2} \\ \frac{1}{2} & 2 \\ 0 & \frac{3}{2} \end{bmatrix}. \]

3.3 Types of Matrices

In this section, we shall discuss different types of matrices.

• (i) Column matrix A matrix is said to be a column matrix if it has only one column. For example,

\[ A = \begin{bmatrix} 0 \\ 3 \\ 1 \\ -1/2 \end{bmatrix} \]

is a column matrix of order 4 × 1. In general, \( A = [a_{ij}]_{m \times 1} \) is a column matrix of order m × 1.

• (ii) Row matrix A matrix is said to be a row matrix if it has only one row. For example,

\[ B = \begin{bmatrix} 1 & 4 & -5 \end{bmatrix} \]

is a row matrix. In general, \( B = [b_{ij}]_{1 \times n} \) is a row matrix of order 1 × n.

• (iii) Square matrix A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For example

\[ A = \begin{bmatrix} 3 & -1 & 0 \\ 3 & 2 & 1 \\ 4 & 3 & -1 \end{bmatrix} \]

is a square matrix of order 3. In general, \( A = [a_{ij}]_{m \times m} \) is a square matrix of order m.

Note If \( A = [a_{ij}] \) is a square matrix of order n, then elements (entries) \( a_{11}, a_{22}, \dots, a_{nn} \) are said to constitute the diagonal, of the matrix A. Thus, if

\[ A = \begin{bmatrix} -1 & 3 & 1 \\ 2 & 4 & -1 \\ 3 & 5 & 6 \end{bmatrix}, \]

Then the elements of the diagonal of A are -1, 4, 6.

• (iv) Diagonal matrix A square matrix B = [b_{ij}]_{m \times m} is said to be a diagonal matrix if all its non diagonal elements are zero, that is a matrix B = [b_{ij}]_{m \times m} is said to be a diagonal matrix if \( b_{ij} = 0 \), when i ≠ j. For example,

\[ A = [4], \quad B = \begin{bmatrix} -1 & 0 \\ 0 & 2 \end{bmatrix}, \quad C = \begin{bmatrix} -1.1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}, \]

are diagonal matrices of order 1, 2, 3, respectively.

• (v) Scalar matrix A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [b_{ij}]_{n \times n} is said to be a scalar matrix if \( b_{ij} = 0 \), when i ≠ j and \( b_{ij} = k \), when i = j, for some constant k. For example

\[ A = [3], \quad B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}, \quad C = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} \]

are scalar matrices of order 1, 2 and 3, respectively.

• (vi) Identity matrix A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. In other words, the square matrix \( A = [a_{ij}]_{n \times n} \) is an identity matrix, if \( a_{ij} = 1 \) if i=j, 0 if i≠j. We denote the identity matrix of order n by I_n. When order is clear from the context, we simply write it as I. For example [1],

\[ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

are identity matrices of order 1, 2 and 3, respectively. Observe that a scalar matrix is an identity matrix when k = 1. But every identity matrix is clearly a scalar matrix.

• (vii) Zero matrix A matrix is said to be zero matrix or null matrix if all its elements are zero. For example, [0],

\[ \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, \quad \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, [0, 0] \]

are all zero matrices. We denote zero matrix by O. Its order will be clear from the context.

3.3.1 Equality of matrices

Definition 2 Two matrices \( A = [a_{ij}] \) and \( B = [b_{ij}] \) are said to be equal if (i) they are of the same order (ii) each element of A is equal to the corresponding element of B, that is \( a_{ij} = b_{ij} \) for all i and j. For example,

\[ \begin{bmatrix} 2 & 3 \\ 0 & 1 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} 2 & 3 \\ 0 & 1 \end{bmatrix} \]

are equal matrices but

\[ \begin{bmatrix} 3 & 2 \\ 2 & 3 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} \]

are not equal matrices. Symbolically, if two matrices A and B are equal, we write A = B.

If

\[ \begin{bmatrix} -1.5 & 0 \\ 2 & 6 \end{bmatrix} = \begin{bmatrix} x & y \\ z & a \end{bmatrix}, \]

then x = – 1.5, y = 0, z = 2, a = 6.

Example 4 (Integrated - Equality Variables)

If \[ \begin{bmatrix} x+3 & z+4 \\ 2y-7 & 3y-2 \end{bmatrix} = \begin{bmatrix} 0 & 6 \\ -3 & 2b+4 \end{bmatrix}, \] find x, y, z, b.

Solution: As the given matrices are equal, therefore, their corresponding elements must be equal. Comparing the corresponding elements, we get x + 3 = 0, z + 4 = 6, 2y – 7 = 3y – 2, a – 1 = – 3, 0 = 2c + 2, b – 3 = 2b + 4. Simplifying, we get a = – 2, b = – 7, c = – 1, x = – 3, y = –5, z = 2.

Example 5 (Integrated - Solve a,b,c,d)

Find the values of a, b, c, and d from the following equation:

\[ \begin{bmatrix} 2+a & 2-b \\ 4-c & 3+d \end{bmatrix} = \begin{bmatrix} 5 & 4 \\ 3 & 11 \end{bmatrix} + \begin{bmatrix} 24 & 0 \\ 0 & 8 \end{bmatrix}. \]

Solution: The right side is \begin{bmatrix} 29 & 4 \\ 3 & 19 \end{bmatrix}. Equating, 2+a=29 ⇒ a=27, 2-b=4 ⇒ b=-2, 4-c=3 ⇒ c=1, 3+d=19 ⇒ d=16.

3.4 Operations on Matrices (Expanded from Book)

Matrix addition and subtraction are defined for matrices of the same order. The sum or difference of two matrices A and B of the same order is the matrix obtained by adding or subtracting the corresponding elements of A and B.

Scalar multiplication: If k is a scalar and A is a matrix, then kA is the matrix obtained by multiplying each element of A by k.

Matrix multiplication: Two matrices A (m × n) and B (n × p) can be multiplied to give a matrix C (m × p) where each element c_ij is the dot product of i-th row of A and j-th column of B.

Properties: Addition is commutative and associative; multiplication is associative and distributive over addition, but not commutative in general.

3.5 Transpose of a Matrix

The transpose of a matrix A, denoted by A^T, is obtained by interchanging rows and columns of A, i.e., \( (A^T)_{ij} = a_{ji} \).

A matrix A is symmetric if A = A^T, skew-symmetric if A = -A^T.

3.6 Symmetric and Skew Symmetric Matrices

Any square matrix can be expressed as sum of symmetric and skew-symmetric matrices: \( A = \frac{A + A^T}{2} + \frac{A - A^T}{2} \).

3.7 Elementary Operation (Transformation) of a Matrix

Row/column operations: Interchange two rows/columns, multiply a row/column by non-zero scalar, add multiple of one row/column to another.

These are used to find inverse or solve systems.

3.8 Inverse of a Matrix

A square matrix A has inverse A^{-1} if AA^{-1} = I = A^{-1}A, exists iff det A ≠ 0.

For 2×2: \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).

General: A^{-1} = \frac{1}{\det A} \adj A, where adj A is transpose of cofactor matrix.

3.9 Solving Linear Equations Using Inverse

For AX = B, X = A^{-1} B if A invertible. Unique solution if det A ≠ 0. Infinite or no solution if singular.

Summary & Exercises Tease

Key Takeaways: Matrices compactly represent linear algebra; operations enable solving systems; inverse crucial for unique solutions. Exercises: Orders/types (3.1), operations (3.2), transpose (3.3), inverse/applications (3.4, Misc).

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

All terms from NCERT Chapter 3, detailed with examples, relevance, proofs where applicable. Grouped by subtopic for easy flashcards. Added advanced terms like "Cramer's Rule Tease" linking to Ch4. Historical notes (Cayley for inverse). Real-life: Google PageRank uses matrices. Use MathJax for book-like rendering.

Tip: Flashcards: Term on front, Ex + Relevance on back. Depth: Prove symmetric decomposition A = (A+A^T)/2 + (A-A^T)/2. Interlinks: Det from Ch4 for inverse. Graphs: Visualize transpose as reflection. Coherent flow: Basics → Types → Ops → Advanced.

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

All 20+ NCERT examples solved in detail, grouped by section. Expanded with steps, diagrams desc, verification. Mirrors book layout with equations rendered.

Tip: Always verify with multiplication back to I or original eq. Practice 2×2 first, then 3×3. For 2025 exams, focus on 3x3 inverses and system solving.

Exercise 3.1 Questions & Answers - Full 10 Qs with MathJax (Book Exact)

Full solutions for all 10 questions on basics/orders/types. Step-by-step like solved examples. Expanded explanations for clarity.

Tip: Systematic: Count rows/cols for order; check zeros for types. Practice constructing from formulas.

Exercise 3.2 Questions & Answers - Full 12 Qs with Proofs

Operations: Addition, multiplication, properties. Full computations with verification.

Tip: Dimensions first; dots for mult. Prove properties element-wise. Verify non-commutativity with examples.

Exercise 3.3 Questions & Answers - Full 12 Qs on Transpose/Symmetric

Transpose properties, symmetric checks. Full with computations and proofs.

Tip: \( A^T_{ij} = A_{ji} \); properties from element definition. Practice decomposition for exams.

Exercise 3.4 Questions & Answers - Full 18 Qs on Minors/Inverses

Minors, cofactors, adjoints, inverses. Full 3×3 computations with sign patterns.

Tip: Sign pattern for cof: + - + ; - + - ; + - +. Verify with det(adj A)=(det A)^{n-1}. Use ops for efficiency in 3x3.

Miscellaneous Exercise Questions & Answers - Full 25 Qs Advanced

Mix: Prove theorems, solve systems, special matrices. Full proofs and calcs. Based on NCERT misc + extensions for practice.

Tip: 6-mark qs: Full inverse calc with verification; proofs using properties. Practice cryptography apps for conceptual.

Quick Revision Notes & Mnemonics (Table Expanded)

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

Subtopic	Key Points	Examples	Mnemonics/Tips
Matrix Basics	Order: m×n, mn elems. Notation [a_ij]. Element: Row i col j. Point: Column vector. Ex: P(0,1)= \begin{bmatrix}0\\1\end{bmatrix}.	Workers 3×2: \begin{bmatrix}30&25\\25&31\\27&26\end{bmatrix} (women III=26).	ORDE: Order=Rows×Dimensions×Elements. Tip: "m Rows, n Cols = mn Total". Pairs for elem count: factors.
Types	CRSDSI Z: Column(m×1), Row(1×n), Square(m=n), Diag(off=0), Scalar(diag=k), Iden(diag=1), Zero(all=0). Diag of sq: a11,a22..	I_3 = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}. Scalar 3I.	CRaZy DiSc Ip ZerO. Tip: Hierarchy Scalar⊂Diag⊂Square. Zero add id, I mult id. Sym A=A^T.
Operations	Add/Sub: Same order, elem-wise. Comm/Assoc. Scalar: k each. Mult: ColA=RowB, dot sum. Assoc, dist add, not comm. AO=O, AI=A.	AB 2×2: \begin{bmatrix}19&22\\43&50\end{bmatrix}. Not BA.	ASMC: Add Scalar Mult Check-dim. Tip: "Add Easy Elem, Mult Dots Careful, Comm Nope!". Dims: m×p.
Transpose/Sym	T: Rows↔Cols, ij=ji. (A+B)^T=A^T+B^T, kA^T=kA^T, (AB)^T=B^TA^T. Sym: A=A^T. Skew: A=-A^T, diag=0. A=Sym+Skew.	Sym \begin{bmatrix}1&2\\2&3\end{bmatrix}. Decomp ex.	TSBA: Trans Swap Back Assoc. Tip: "Flip for T, Self for Sym, Neg for Skew". Decomp always works.
Inverse/Apps	Inv: det≠0, A^{-1}=adj/det. adj=C^T, C=(-1)^{i+j}M. M=det submatrix. AX=B: X=A^{-1}B unique. Elem ops: Swap, scale, add mult. (AB)^{-1}=B^{-1}A^{-1}.	2×2: (1/det)\begin{bmatrix}d&-b\\-c&a\end{bmatrix}. 3×3 ex det=1.	IMCA: Inverse Minor Cof Adjoint. Tip: "Det≠0 first, Sign +-, Adjoint Trans Cof, Apps Decode/Systems". Gauss ops fast.

Overall Mnemonic: ORDE-CRSDSI-ASMC-TSBA-IMCA (scan 3 mins). Flashcards: 1 per row. 2025 Exam: 3×3 inv (4 marks), system solve (6 marks), proofs (2 marks).