NCERT Class 12 Maths Chapter 3 Notes, Matrices Class 12 Chapter 3 Notes

Do you know how to solve a complex system of equations? How do search engines like Google list the websites one by one? Or, how is the congested traffic flow managed in busy streets? The answers to all these questions can be found in Matrices, a powerful mathematical tool that can be used to solve complex systems of equations, also manage and manipulate data efficiently. From NCERT Class 12 Maths, the chapter Matrices, contains the Definition of a Matrix, Order of a Matrix, Types of Matrices, Operations on Matrices, Transpose of a Matrix, etc. Understanding these concepts will enable the students to solve matrix-related problems easily and also enhance their problem-solving ability in real-world applications.

This article on NCERT notes Class 12 Maths Chapter 3 Matrices offers well-structured NCERT Notes to help the students grasp the concepts of Matrices easily. Students who want to revise the key topics of Matrices quickly will find this article very useful. It will also boost the exam preparation of the students by many folds. These NCERT Class 12 Maths Notes are made by the Subject Matter Experts according to the latest CBSE syllabus, ensuring that students can grasp the basic concepts effectively. NCERT solutions for class 12 maths and NCERT solutions for other subjects and classes can be downloaded from the NCERT Solutions.

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NCERT Notes for Class 12 Chapter 3 Matrices:

Matrices are powerful mathematical tools used to simplify the process of solving systems of linear equations and have many other applications far beyond mathematics. They are mostly used in fields of computer science, physics, economics, cryptography, and even social sciences for data analysis, data transformations, and modelling.

Matrix:

A rectangular arrangement of objects (numbers or symbols, or any other objects) is called a matrix (plural: matrices).
Examples:

1. $\left[\begin{array}{ccc}2& 4& -3\\ 5& 4& 6\end{array}\right]$
2. $\left[\begin{array}{cc}2& 4i+3\\ 5& 4\\ 3i& -75\end{array}\right]$

Order of a Matrix:

Rows and Columns:
The horizontal objects denote a row, and the vertical ones denote a column.
E.g., in the first matrix above, elements 2, 4 and -3 lie in the first row and 5, 4 and 6 in the second row.
Also, 2, 5 lie in the first column, 4,4 in the second column, and -3 and 6 in the third column.

Order of a matrix:
A matrix of order m × n (read as m by n matrix) means that the matrix has $m$ rows and $n$ columns.
E.g., the first matrix has order 2 x 3.
The second matrix has order 3 x 2.
The third matrix has order 4 x 1.

Representation of a $m\times n$ matrix:
$\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ \dots & \dots & \dots & \dots \\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]$
This representation can be represented in a more compact form as ${\left[a_{ij}\right]}_{m\times n}$
Where $a_{ij}$ represents the element of ith row and jth column and i = 1,2,...,m; j = 1,2,...,n.
For example, to locate the entry in matrix A identified as aij, we look for the entry in row i, column j. In matrix A, shown below, the entry in row 2, column 3 is a23.
$A=\left[\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$
Note: A matrix is only a representation of the symbol, number, or object. It does not have any value. Usually, a matrix is denoted by capital letters.

Types of Matrices

Row Matrix (Row vector) :
A matrix that has only one row.
$\mathrm{A}=[{\mathrm{a}}_{11},{\mathrm{a}}_{12},\dots \dots \dots \dots {\mathrm{a}}_{1\mathrm{n}}]$

Column Matrix (Column vector) :
A matrix that has only one column.
$A=\left[\begin{array}{c}{a}_{11}\\ a_{21}\\ .\\ .\\ .\\ a_{m1}\end{array}\right]$

Zero or Null Matrix :
A matrix whose all entries are zero.
$\mathrm{A}=\left[\begin{array}{ll}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

Square Matrix :
In a square matrix, the number of rows is equal to the number of columns.
$A=\left[\begin{array}{rrr}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]$

Diagonal matrix:
A square matrix is said to be a diagonal matrix if all its elements except the diagonal elements are zero.
So, a matrix $\mathrm{A}={\left[{\mathrm{a}}_{\mathrm{ij}}\right]}_{\mathrm{m}\times \mathrm{n}}$ is a diagonal matrix if ${\mathrm{a}}_{\mathrm{ij}}=0$, whenever $\mathrm{i}\ne \mathrm{j}$ and $\mathrm{m}=\mathrm{n}$.
E.g
$\left[\begin{array}{ccc}{a}_{11}& 0& 0\\ 0& a_{22}& 0\\ 0& 0& a_{33}\end{array}\right]$
A diagonal matrix of order n x n having diagonal elements as d1, d2, d3 ………, dn is denoted by $\mathrm{diag}[d_1,d_2,d_3\dots \dots \dots ,d_n]$
For example:
$A=\left[\begin{array}{cc}6& 0\\ 0& -7\end{array}\right]B=\left[\begin{array}{ccc}2& 0& 0\\ 0& -9& 0\\ 0& 0& 3\end{array}\right]$
So, we can write
$\mathrm{A}=\mathrm{diag}[6,-7]\text{ and }\mathrm{B}=\mathrm{diag}[2,-9,3]$

Scalar matrix:
A diagonal matrix whose all diagonal elements are equal is called a scalar matrix.
$\mathrm{A}=\left[\begin{array}{ll}3& 0\\ 0& 3\end{array}\right]\mathrm{B}=\left[\begin{array}{ccc}-3& 0& 0\\ 0& -3& 0\\ 0& 0& -3\end{array}\right]$
For a square matrix $\mathrm{A}={\left[{\mathrm{a}}_{\mathrm{ij}}\right]}_{\mathrm{n}\times \mathrm{n}}$ to be scalar matrix
${\mathrm{a}}_{\mathrm{ij}}=\{\begin{array}{ll}0,& i\ne j\\ c,& i=j\end{array}$
Where c is not equal to 0

Unit or Identity Matrix:
A diagonal matrix of order n whose all diagonal elements are equal to one is called an identity matrix of order n. It is represented as I.
So, a square matrix $A={\left[a_{ij}\right]}_{n\times n}$ is Identity matrix if

${\mathrm{a}}_{\mathrm{ij}}=\{\begin{array}{ll}0,& i\ne j\\ 1,& i=j\end{array}$
For example,
${\mathrm{I}}_3=\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Equality of Matrices:

Matrices A and B are equal if '
(i) The order of matrix A is equal to the order of matrix B.
(ii) $a_{ij}=b_{ij}$ for each pair

Operations on Matrices

In this part, we will learn about operations on matrices, namely, the addition of matrices, the multiplication of a matrix by a scalar, the difference and multiplication of matrices

Addition of Matrices:

If matrices A and B are of the same order, then the addition of matrices A and B is given by
A + B = [aij + bij]

Properties:
(a) Addition of matrices is commutative: i.e. A + B = B + A
(b) Matrix addition is associative : (A + B) + C = A + (B + C)

Additive inverse: Let two matrices A and B be such that A + B = O = B + A, then B is called the additive inverse of A.

Multiplication of a Matrix by a Scalar:

If A and B are two matrices of the same order and 'k' is a scalar, then k(A + B) = kA + kB.
If p and q are two scalars and 'A' is a matrix, then (p+q )A = pA + qA
If p and q are two scalars and 'A' is a matrix, then (pq )A = p(qA) = q(pA)

Multiplication of Matrices:

Let A be a matrix of order m x n and B be a matrix of order p x q, then the matrix multiplication AB is possible if and only if n = p. The multiplication of AB is given by

$(\mathrm{AB}{)}_{ij}=\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{ir}}{\mathrm{b}}_{\mathrm{rj}}$

Properties of Matrix Multiplication:

(i) Matrix multiplication is not commutative: i.e. $AB\ne BA$

( ii ) Matrix Multiplication Is Associative: If A, B & C are conformable for the product AB & BC, then (AB) C = A(BC)

( ii ) Matrix Multiplication Is Distributivity : A(B+C)=A B+A C and (A+B) C=A C+B C

Transpose of a Matrix:

Let $A=[a_{ij}]$be any matrix of order m x n. Then transpose of matrix A i.e. AT or $A^{\prime }=[a_{ji}]$

Properties of transpose:

• $(\mathrm{A}+\mathrm{B}{)}^{\mathrm{T}}={\mathrm{A}}^{\mathrm{T}}+{\mathrm{B}}^{\mathrm{T}}$

• $(\mathrm{A}\mathrm{B}{)}^{\mathrm{T}}={\mathrm{B}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}$

• ${\left({\mathrm{A}}^{\mathrm{T}}\right)}^{\mathrm{T}}=\mathrm{A}$

• $(\mathrm{kA}{)}^{\mathrm{T}}={\mathrm{kA}}^{\mathrm{T}}$where 'k' is a scaler

Orthogonal Matrix:

A square matrix is said to be an orthogonal matrix if
$\mathrm{A}{\mathrm{A}}^{\mathrm{T}}=\mathrm{I}$

Symmetric and Skew-Symmetric Matrices:

A square matrix $A=[aij]$ is said to be a symmetric matrix if $a_{ij}=a_{ji}$ for each element in the matrix

A square matrix $A=[aij]$ is said to be a skew-symmetric matrix if $a_{ij}=-a_{ji}$ for each element in the matrix

Properties of symmetric & skew-symmetric matrices:

1. Matrix A is symmetric if $A^T=A$

2. Matrix A is skew-symmetric if $A^T=-A$

3. The sum of two symmetric matrices is a symmetric matrix, and the sum of two skew-symmetric matrices is a skew-symmetric matrix.

4. If A & B are symmetric matrices, then AB + BA is a symmetric matrix and AB – BA is a skew-symmetric matrix.

Elementary operations of a matrix:

The following elementary operations are allowed on a matrix.

• ${\mathrm{R}}_{\mathrm{i}}\leftrightarrow {\mathrm{R}}_{\mathrm{j}}or{\mathrm{C}}_{\mathrm{i}}\leftrightarrow {\mathrm{C}}_{\mathrm{j}}$

• ${\mathrm{R}}_i\to k{\mathrm{R}}_{i}or{\mathrm{C}}_i\to k{\mathrm{C}}_i$

• ${\mathrm{R}}_i\to {\mathrm{R}}_{\mathrm{j}}+k{\mathrm{R}}_{i}or{\mathrm{C}}_i\to {\mathrm{C}}_i+k{\mathrm{C}}_j$

Inverse of a Matrix:

If A and B are two square matrices such that $AB=BA=I$, then B is the inverse matrix of A and is denoted by A^–1, and A is the inverse of B.

Note:

• It is necessary that matrices A and B should be square matrices of the same order to be invertible.
• If B is the inverse of A, then A is also the inverse of B.

Matrices: Previous Year Question and Answer

Given below are some previous year question answers of various examinations from the NCERT class 12 chapter 3, Matrices:

Question 1: If $\left[\begin{array}{cc}x+y& 3y\\ 3x& x+3\end{array}\right]=\left[\begin{array}{cc}9& 4x+y\\ x+6& y\end{array}\right]$, then $(x-y)=$ ?

Solution:
$\left[\begin{array}{cc}x+y& 3y\\ 3x& x+3\end{array}\right]=\left[\begin{array}{cc}9& 4x+y\\ x+6& y\end{array}\right]$
Equating matrix elements gives:
$x+y=9\dots (1)$
$3y=4x+y⟹2y=4x⟹y=2x\dots (2)$
$3x=x+6⟹2x=6⟹x=3\dots (3)$
$x+3=y\dots (4)$
From (3), $x=3$.
Putting the value of $x=3$ in (2)
$y=2(3)=6$
Now, $x-y=3-6$
⇒ $x-y=-3$
Hence, the correct answer is $-3$.

Question 2: Let $A$ be a matrix of order $m\times n$ and $B$ be a matrix such that $A^{T}B$ and $B{A}^T$ are defined. Then, the order of $B$ is:

Solution:
Order of matrix $\mathrm{A}=\mathrm{m}\times \mathrm{n}$
Let order of matrix B be $\mathrm{K}\times \mathrm{P}$
Order of matrix ${\mathrm{B}}^T=\mathrm{P}\times \mathrm{K}$
If $A{B}^T$ is defined then the order of $A{B}^T$ is $m\times K$ if $n=P$
If ${\mathrm{B}}^T\mathrm{A}$ is defined then order of ${\mathrm{B}}^T\mathrm{A}$ is $\mathrm{P}\times \mathrm{n}$ when $\mathrm{K}=\mathrm{m}$
Now, order of ${\mathrm{B}}^T=\mathrm{P}\times \mathrm{K}$
$\therefore$ Order of $\mathrm{B}=\mathrm{K}\times \mathrm{P}$
$=\mathrm{m}\times \mathrm{n}\dots ..[\because \mathrm{K}=\mathrm{m},\mathrm{P}=\mathrm{n}]$
Hence, the correct answer is $\mathrm{m}\times \mathrm{n}$.

Question 3: If $A$ is a square matrix of order 3 such that $\det (A)=9$, then $\det \left(9A^{-1}\right)$ is equal to:

Solution:
Given $\det (A)=9$ for a $3\times 3$ matrix $A$.
Using determinant properties:
$\det (kA)=k^n\det (A)$
$\det (A^{-1})=\frac{1}{\det (A)}$
$\det (9A^{-1})=9^3\det (A^{-1})$
$=9^3\cdot \frac{1}{\det (A)}$
$=9^3\cdot \frac{1}{9}$
$=9^2$
Hence, the correct answer is $9^2$.

NCERT Class 12 Notes Chapter Wise

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Subject-Wise NCERT Exemplar Solutions

• NCERT Exemplar Class 12 Solutions
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• NCERT Exemplar Class 12 Physics
• NCERT Exemplar Class 12 Chemistry
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• NCERT Solutions for Class 12 Mathematics
• NCERT Solutions for Class 12 Chemistry
• NCERT Solutions for Class 12 Physics
• NCERT Solutions for Class 12 Biology

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