Matrices Class 12th Notes - Free NCERT Class 12 Maths Chapter 3 Notes - Download PDF

NCERT Class 12 Maths Chapter 3 Matrices Notes- Matrices was introduced to solve simultaneous linear equations. It has an important application in most scientific fields like classical mechanics, optics, electromagnetism, quantum mechanics, quantum electrodynamics and, many more concepts of physics. In this article, you will get NCERT Class 12 Maths Chapter 3 Notes which are very important for the CBSE board exam as well as competitive exams. These notes are prepared by experts who know how best to prepare for board exams. Students are advised to solve all the NCERT problems on their own. You can take help from Matrices Class 12 Notes, which will give a short description of the topics covered in this chapter. Here, you can also check NCERT Book

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• NCERT Notes for Class 12 Maths
• NCERT Solutions for Class 12 Maths Chapter 3 Matrices
• NCERT Exemplar Class 12 Maths Solutions Chapter 3 Matrices

Matrix:-

A rectangular array of numbers horizontal lines and vertical lines is called a matrix. Let's there are 'm' horizontal lines and 'n' vertical lines then the order of the matrix is m by n which is written as m x n matrix.

$A=\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{\ln }\\ a_{21}& a_{22}& a_{23}& \dots & a_{2n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{m1}& a_{m2}& a_{m3}& \dots & a_{mn}\end{array}\right]$ aij belongs to the ith row and jth column and is called the (i, j) th element of the matrix

$A=\left[\begin{array}{ccc}1& 2& 3\\ 0& -1& 9\end{array}\right]$

$a_{11}=1,a_{12}=2,a_{13}=3,a_{21}=0,a_{22}=-1,a_{23}=9$

SPECIAL TYPE OF MATRICES :

Row Matrix (Row vector) :

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) :

Matrix that has only one column.

$A=\left[\begin{array}{c}{a}_{11}\\ a_{21}\\ .\\ .\\ .\\ a_{m1}\end{array}\right]$

Zero or Null Matrix :

Matrix whose all entries are zero.

$\mathrm{A}=\left[\begin{array}{ll}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

Horizontal Matrix:

A matrix that has more columns than the number of rows.

$\left[\begin{array}{llll}1& 2& 3& 4\\ 2& 5& 1& 1\end{array}\right]$

Vertical Matrix :

A matrix that has more rows than the number of columns.

$A=\left[\begin{array}{ll}2& 5\\ 1& 1\\ 3& 6\\ 2& 4\end{array}\right]$

Square Matrix :

In a square matrix number of rows are equal to the number of columns.

$A=\left[\begin{array}{rrr}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]$

Trace of Matrix :

The sum of the elements of a square matrix A lying along the principal diagonal is called the trace of A.

$A=\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{\ln }\\ a_{21}& a_{22}& a_{23}& \dots & a_{2n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{n1}& a_{n2}& a_{n3}& \dots & a_{nn}\end{array}\right]$

$\mathrm{tr}(\mathrm{A})=\sum _{i=1}^{\mathrm{a}}{\mathrm{a}}_{\mathrm{ii}}={\mathrm{a}}_{11}+{\mathrm{a}}_{22}+\dots \dots {\mathrm{a}}_{\mathrm{nn}}$

Properties of trace of a matrix :

Let A and B are two matrices.

$$\begin{gathered} (i)\mathrm{tr}(\lambda ,A)=\lambda \mathrm{tr}(A) \\ (ii)\mathrm{tr}(\mathrm{A}+\mathrm{B})=\mathrm{tr}(A)+tr(B) \\ (iii)\mathrm{tr}(\mathrm{AB})=\mathrm{tr}(\mathrm{BA}) \end{gathered}$$

Equality of Matrices:

Matrices A and B are equal if '

(i) Order of matrix A are equal to order of matrix B.

(ii) $a_{ij}=b_{ij}$ for each pair

Addition of Matrices:

If matrix A and matrix are of same order then addition of matix A and B 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 are such that A + B = O = B + A, then B is called additive inverse of A.

Multiplication of Matrix by a Scalar:

- If A and B are two matrices of the same order and 'k' be 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 Matrix:

Let $A=[a_{ij}]$be any matrix of order m x n. Then transpose of matrix A i.e. A^T 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 matrix :

A square matrix $A=[aij]$ is said to be a symmetric matrix if

$a_{ij}=a_{ji}$ for each element in the matrix

Skew symmetric 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 matrix :

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:

Following elementary operations are allowed on 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 matrices.

- If B is the inverse of A, then A is also the inverse of B

NCERT Class 12 Notes Chapter Wise.

Subject Wise NCERT Exemplar Solutions

• NCERT Exemplar Class 12 Solutions
• NCERT Exemplar Class 12 Maths
• NCERT Exemplar Class 12 Physics
• NCERT Exemplar Class 12 Chemistry
• NCERT Exemplar Class 12 Biology

Subject Wise NCERT Solutions

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

NCERT Books and Syllabus

Important points to Note:-

• NCERT problems are very important in order to perform well in the exam. Students must solve all the NCERT problems including miscellaneous exercises on their own.

• Students are advised to go through NCERT Class 12 Maths Chapter 3 Notes before solving the questions.

• You can take help from these Matrices Class 12 Notes for quick revision before the exam.

Happy learning !!!