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Class 12 · Maths · Chapter 3 · NCERT Class 12 Mathematics Part 1

Matrices Class 12 Notes

Free: the full mind map and notes for all 7 parts of the chapter. Checking your answers is free with an account.

Chapter mind map

The whole chapter at a glance: the big idea, then each branch and what sits under it.

Matrices

A rectangular array of numbers or functions used to organize data and solve linear systems in physics, economics, and computer science.

  1. Matrix Fundamentals

    Basic structure, notation, and the concept of matrix equality.

    • Definition and Notation — An ordered rectangular array of elements; A = [a_ij] denotes the element at the i-th row and j-th column.
    • Order of a Matrix — An m x n matrix has m rows and n columns; total elements equal the product mn.
    • Equality of Matrices — Two matrices are equal if they have the same order and every corresponding element is identical (a_ij = b_ij).

  2. Classification of Matrices

    Types of matrices based on the arrangement and values of their elements.

    • Row and Column Matrices — Column matrix has only one column (m x 1); Row matrix has only one row (1 x n).
    • Square and Diagonal Types — Square (m=n); Diagonal (non-diagonals are 0); Scalar (diagonal elements equal k); Identity (diagonal elements are 1).
    • Zero Matrix — A matrix (O) where every element is zero; serves as the additive identity in matrix addition.

  3. Matrix Operations

    Rules for addition, scalar multiplication, and the complex process of matrix multiplication.

    • Addition and Subtraction — Defined for same-order matrices; commutative and associative. A - B is defined as A + (-B).
    • Scalar Multiplication — Multiplying every element by a constant k. The negative of a matrix is (-1)A.
    • Multiplication Compatibility — AB is defined only if columns of A equal rows of B. Resulting order is m x p for (m x n) and (n x p) matrices.
    • Multiplication Properties — Generally non-commutative (AB != BA); however, it is associative and distributive over addition.

  4. Transpose and Symmetry

    Operations involving row-column swapping and resulting matrix symmetries.

    • Transpose Properties — Formed by swapping rows and columns. Key laws: (A')' = A, (kA)' = kA', and Reversal Law (AB)' = B'A'.
    • Symmetric Matrices — A square matrix where A' = A; elements are mirrored across the main diagonal (a_ij = a_ji).
    • Skew Symmetric Matrices — A square matrix where A' = -A; diagonal elements must be zero because a_ii = -a_ii.
    • Matrix Decomposition Theorem — Any square matrix A can be uniquely expressed as the sum of a symmetric matrix 1/2(A+A') and a skew-symmetric matrix 1/2(A-A').

  5. Invertible Matrices

    The matrix equivalent of division, applicable to specific square matrices.

    • Inverse Definition — If AB = BA = I for square matrices A and B, then B is the unique inverse A^-1.
    • Existence and Uniqueness — Only square matrices can be invertible; rectangular matrices cannot. If an inverse exists, it is unique.
    • Inverse Reversal Law — For invertible matrices A and B of the same order, (AB)^-1 = B^-1 A^-1.

Chapter notes

A comprehensive study of matrices, including their definitions, types, operations, and properties such as symmetry and invertibility, based on the Class 12 Mathematics curriculum.

01 · Explore

Introduction and Definition of a Matrix

A matrix is a powerful mathematical tool used to organize data and solve systems of linear equations. It has applications in physics, economics, genetics, and computer science.

Formally, a matrix is an ordered rectangular array of numbers or functions. These numbers or functions are called the elements or entries of the matrix. We typically denote matrices with capital letters like A, B, or C.

The horizontal lines of elements are called rows, and the vertical lines are called columns. For example, if a matrix has 3 rows and 2 columns, it is said to have an order of 3 × 2 (read as '3 by 2').

In general, an m × n matrix has m rows and n columns. The element in the i-th row and j-th column is denoted as a_ij. The total number of elements in such a matrix is the product mn.

Matrix NotationDescription
A = [a_ij]_m x nCompact notation for a matrix with m rows and n columns.
a_ijThe element located at the intersection of the i-th row and j-th column.
Order m x nSpecifies the dimensions (rows by columns) of the matrix.

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If a matrix has 12 elements, what are all the possible orders it can have?

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NCERT reference: chapter PDF pages 1, 2, 3.

02 · Explore

Types of Matrices

Matrices are classified based on the arrangement and values of their elements.

A Column Matrix has only one column (order m × 1), while a Row Matrix has only one row (order 1 × n). A Square Matrix is one where the number of rows equals the number of columns (m = n).

Within square matrices, a Diagonal Matrix has all non-diagonal elements as zero. If all diagonal elements of a diagonal matrix are equal to a constant 'k', it is a Scalar Matrix. If that constant is 1, it is an Identity Matrix (denoted by I).

A Zero Matrix (or null matrix) is a matrix where every single element is zero. It is denoted by O. Equality of matrices occurs only if two matrices have the same order and every corresponding element is identical (a_ij = b_ij).

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What is the difference between a scalar matrix and an identity matrix?

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NCERT reference: chapter PDF pages 6, 7, 8.

03 · Explore

Matrix Addition and Scalar Multiplication

Basic operations like addition and subtraction can be performed on matrices, provided they meet specific dimensional requirements.

Addition of matrices is defined only when the matrices are of the same order. To add two matrices A and B, we simply add their corresponding elements: C = [a_ij + b_ij].

Scalar multiplication involves multiplying every element of a matrix by a constant 'k'. For example, if A = [2, 4], then 3A = [6, 12]. The negative of a matrix, -A, is simply (-1) multiplied by A.

The difference A - B is calculated as A + (-B). Matrix addition follows the Commutative Law (A + B = B + A) and the Associative Law ((A + B) + C = A + (B + C)). The zero matrix O acts as the additive identity (A + O = A).

Finding a Matrix in an Equation

2A + 3X = 5B, where A = [[8, 0], [4, -2], [3, 6]] and B = [[2, -2], [4, 2], [-5, 1]]

1. Rearrange for X: 3X = 5B - 2A. 2. Calculate 5B: [[10, -10], [20, 10], [-25, 5]]. 3. Calculate 2A: [[16, 0], [8, -4], [6, 12]]. 4. Subtract: 5B - 2A = [[-6, -10], [12, 14], [-31, -7]]. 5. Divide by 3: X = [[-2, -10/3], [4, 14/3], [-31/3, -7/3]].

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Can you add a 2x3 matrix to a 3x2 matrix?

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NCERT reference: chapter PDF pages 10, 11, 12, 13, 14.

04 · Explore

Multiplication of Matrices

Matrix multiplication is a more complex operation that involves a row-by-column dot product.

The product of two matrices A and B is defined only if the number of columns in A is equal to the number of rows in B. If A is of order m × n and B is of order n × p, the resulting matrix C will be of order m × p.

To find the element c_ik, we take the i-th row of A and the k-th column of B, multiply corresponding elements, and sum them up. Formally, c_ik = Σ (a_ij * b_jk) for j = 1 to n. This is why the 'inner' dimensions must match.

Crucially, matrix multiplication is generally NOT commutative (AB ≠ BA). Even if both products are defined, they may have different orders or different elements. However, it is associative (A(BC) = (AB)C) and distributive over addition.

Matrix Multiplication Process

  1. 1

    Check Dimensions

    Ensure columns of Matrix A equal rows of Matrix B.

  2. 2

    Select Row and Column

    Pick the i-th row from A and the k-th column from B.

  3. 3

    Multiply and Sum

    Multiply corresponding elements and add them to find entry c_ik.

  4. 4

    Repeat

    Perform this for every row in A and every column in B.

The sequential steps to compute the product of two matrices.

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If A is a 2x3 matrix and B is a 3x4 matrix, what is the order of AB?

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NCERT reference: chapter PDF pages 17, 18, 19, 20, 21, 22, 23, 24, 25.

05 · Explore

Transpose of a Matrix

The transpose of a matrix is formed by swapping its rows and columns.

If A = [a_ij] is an m × n matrix, then the transpose of A, denoted by A' or Aᵀ, is the n × m matrix [a_ji]. Effectively, the first row of A becomes the first column of A', the second row becomes the second column, and so on.

Properties of transpose include: (i) (A')' = A, (ii) (kA)' = kA', (iii) (A + B)' = A' + B', and (iv) (AB)' = B'A'. The last property is known as the Reversal Law of Transpose.

Transposing is a fundamental operation used to define symmetric and skew-symmetric matrices.

Verifying (AB)' = B'A'

A = [[-2], [4], [5]], B = [[1, 3, -6]]

1. AB = [[-2, -6, 12], [4, 12, -24], [5, 15, -30]]. 2. (AB)' = [[-2, 4, 5], [-6, 12, 15], [12, -24, -30]]. 3. A' = [[-2, 4, 5]], B' = [[1], [3], [-6]]. 4. B'A' = [[1*-2, 1*4, 1*5], [3*-2, 3*4, 3*5], [-6*-2, -6*4, -6*5]] = [[-2, 4, 5], [-6, 12, 15], [12, -24, -30]]. Result: (AB)' = B'A'.

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What is the transpose of a 1x3 row matrix?

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NCERT reference: chapter PDF pages 28, 29.

06 · Explore

Symmetric and Skew Symmetric Matrices

These are special types of square matrices defined by their relationship with their transpose.

A square matrix A is Symmetric if A' = A. This means the elements are mirrored across the main diagonal (a_ij = a_ji).

A square matrix A is Skew Symmetric if A' = -A. This implies a_ji = -a_ij. For the diagonal elements where i = j, this means a_ii = -a_ii, which forces all diagonal elements of a skew-symmetric matrix to be zero.

Theorem: Any square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix. Specifically, A = (1/2)(A + A') + (1/2)(A - A'), where (A + A') is symmetric and (A - A') is skew-symmetric.

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Why must the diagonal elements of a skew-symmetric matrix be zero?

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NCERT reference: chapter PDF pages 30, 31, 32, 33.

07 · Explore

Invertible Matrices

Inversion is the matrix equivalent of division for real numbers, applicable only to square matrices.

If A is a square matrix of order m, and there exists another square matrix B of the same order such that AB = BA = I, then B is called the inverse of A (denoted as A⁻¹). In this case, A is said to be invertible.

A rectangular matrix cannot have an inverse because for the products AB and BA to be defined and equal to the identity matrix, the matrices must be square and of the same order. Furthermore, the inverse of a square matrix, if it exists, is unique.

A key property of invertible matrices is (AB)⁻¹ = B⁻¹A⁻¹. This reversal law also applies to the transpose of a product: (AB)' = B'A'.

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If B is the inverse of A, what is the inverse of B?

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Can a 2x3 matrix have an inverse?

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NCERT reference: chapter PDF pages 35, 36.

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