# NCERT Solutions for Class 12 Maths Chapter 3

Free NCERT Solutions for Class 12 Maths Chapter 3 Matrices in English and Hindi Medium for CBSE and UP Board students in PDF form. Download CBSE Apps based on NCERT Solutions to prepare yourself according to latest CBSE Syllabus 2019-20 for the new session.

 Class: 12 Subject: Maths – गणित Chapter 3: Matrices

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## NCERT Solutions for Class 12 Maths Chapter 3

Download NCERT Solutions Class 12 maths chapter 3 exercise 3.1, 3.2, 3.3, 3.4 and miscellaneous exercises in PDF form English and Hindi Medium for CBSE & UP Board students following NCERT Books for the academic session 2019-20. Alignment of some questions may change, if you feel some problem, inform us. We will rectify.

### Class 12 Maths Chapter 3 Matrices Solutions – English & Hindi Medium

• 12 Maths Chapter 3 Exercise 3.1 Solutions
• 12 Maths Chapter 3 Exercise 3.2 Solutions
• 12 Maths Chapter 3 Exercise 3.3 Solutions
• 12 Maths Chapter 3 Exercise 3.4 Solutions
• 12 Maths Chapter 3 Miscellaneous Exercise 3 Solutions

### Assignments for practice

#### Mixed Chapter Tests

Chapter 1, 2, 3 & 4

##### Level 3 Test 1 Test 2

Matrix

The arrangement of real numbers in a rectangular array enclosed in brackets as [] or () is known as a Matrix(Matrices is plural of matrix). Matrix operations are used in electronic physics, computers, budgeting, cost estimation, analysis and experiments. They are also used in cryptography, modern psychology, genetics, industrial management etc. In general an m x n matrix is matrix having m rows and n columns. it can be written as follows:

Order of a Matrix
There may be any number of rows and any number of columns in a matrix. If there are m rows and n columns in matrix A, its order is m x n and it is read as an m x n matrix.

Transpose of a Matrix
The transpose of a given matrix A is formed by interchanging its rows and columns and is denoted by A’.
Symmetric Matrix
A square matrix A is said to be a symmetric matrix if A’ = A.
Skew-Symmetric Matrix
A square matrix A is said to be a skew symmetric if A’ = – A. all elements in the principal diagonal of a skew symmetric matrix are zeroes.
If A and B are any two given matrices of the same order, then their sum is defined to be a matrix C whose respective elements are the sum of the corresponding elements of the matrices A and B and we write this as C = A + B.

##### Type of Matrices

Row matrix: A row matrix has only one row but any number of columns.
Column matrix: A column matrix has only one column but any number of rows.
Square matrix: A square matrix has the number of column equal to the number of rows.
Rectangular Matrix: A matrix is said to be a rectangular matrix if the number of rows is not equal to the number of columns.

Diagonal matrix: If in a square matrix has all elements 0 except principal diagonal elements, it is called diagonal matrix.

Scalar Matrix: A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant.
Zero or Null matrix: If all elements of a matrix are zero, then the matrix is known as zero matrix and denoted by O.

Unit or Identity matrix: If in a square matrix has all elements 0 and each diagonal elements are non-zero, it is called identity matrix and denoted by I.
Equal Matrices: Two matrices are said to be equal if they are of the same order and if their corresponding elements are equal.

##### Properties of Matrix
• When a matrix is multiplied by a scalar, then each of its element is multiplied by the same scalar.
• If A and B are any two given matrices of the same order, then their sum is defined to be a matrix C whose respective elements are the sum of the corresponding elements of the matrices A and B and we write this as C = A + B.
• For any two matrices A and B of the same order, A + B = B + A. i.e. matrix addition is commutative.
• For any three matrices A, B and C of the same order, A + (B + C) = (A + B) + C i.e., matrix addition is associative.
• Additive identity is a zero matrix, which when added to a given matrix, gives the same given matrix, i.e., A + O = A = O + A.
• If A + B = O, then the matrix B is called the additive inverse of the matrix of A.
• If A and B are two matrices of order m x p and p x n respectively, then their product will be a matrix C of order m x n.

###### Invertible Matrix

A square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I = BA, Where I is identify matrix of order n.
Theorems of invertible matrices

• Theorem 1: Every invertible matrix possesses a unique inverse.
• Theorem 2: A square matrix is invertible iff it is non-singular.

Historical Facts!
Matrix is a latin word. Originally matrices are used for solutions of simultaneous linear equations. An important Chinese Text between 300 BC and 200 AD, nine chapters of Mathematical Art(Chiu Chang Suan Shu), give the use of matrix methods to solve simultaneous equations. Carl Friedrich Gauss(1777 – 1855) also gave the method to solve simultaneous equations by matrix method.

#### If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

The number of elements = 24
Therefore, the possible orders are as follows: 1×24,2×12,3×8,4×6,6×4,8×3,12×2 and 24×1
If it has 13 elements, then the possible orders: 13×1 and 1×13

#### यदि किसी आव्यूह में 18 अवयव हैं तो इसकी संभव कोटियाँ क्या हैं? यदि इसमें 5 अवयव हों तो क्या होगा?

आव्यूह में कुल अवयव = 18
इसलिए, आव्यूह की संभव कोटियाँ निम्नलिखित हैं:
1×18,2×9,3×6,6×3,9×2 और 18×1
यदि इसमें 5 अवयव हों तो कोटियाँ निम्नलिखित होंगी: 5×1 और 1×5

#### If A, B are symmetric matrices of same order, then what type of matrix AB – BA is?

(𝐴𝐵 − 𝐵𝐴)′=(𝐴𝐵)′ − (𝐵𝐴)′ [∵ (𝑋 − 𝑌)′=𝑋′−𝑌′] =𝐵′𝐴′ − 𝐴′𝐵′ [∵ (𝑋𝑌)′=𝑌′𝑋′] =𝐵𝐴 − 𝐴𝐵 [∵ Given: 𝐴′=𝐴,𝐵′=𝐵] =− (𝐴𝐵−𝐵𝐴)
⇒(𝐴𝐵 − 𝐵𝐴)′
=−(𝐴𝐵 − 𝐵𝐴),
Therefore,the matrix (AB − BA) is a skew symmetric matrix.

#### Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

If 𝐴 is a symmetric matrix, then 𝐴′=𝐴
Here, (𝐵′𝐴𝐵)′= (𝐴𝐵)′(𝐵′)′ [∵ (𝐴𝐵)′=𝐵′𝐴′] =(𝐴𝐵)′𝐵 [∵ (𝐵′)′ =𝐵] =𝐵′𝐴′𝐵 [∵ (𝐴𝐵)′=𝐵′𝐴′] =𝐵′𝐴𝐵 [∵ Given 𝐴′=𝐴] ⇒(𝐵′𝐴𝐵)′=𝐵′𝐴𝐵,
Hence,the matrix 𝐵′𝐴𝐵 is also symmetric.
If 𝐴 is skew symmetric matrix, then 𝐴′=−𝐴
Here, (𝐵′𝐴𝐵)′= (𝐴𝐵)′(𝐵′)′ [∵ (𝐴𝐵)′=𝐵′𝐴′] =(𝐴𝐵)′𝐵 [∵ (𝐵′)′ =𝐵] =𝐵′𝐴′𝐵 [∵ (𝐴𝐵)′=𝐵′𝐴′] =−𝐵′𝐴𝐵 [∵ Given 𝐴′=−𝐴] ⇒(𝐵′𝐴𝐵)′=−𝐵′𝐴𝐵,
Hence,the matrix 𝐵′𝐴𝐵 is also a skew-symmetric matrix.

#### यदि A तथा B सममित आव्यूह हैं तो सिद्ध कीजिए कि AB-BA एक विषम सममित आव्यूह है।

(AB – BA)^’=(AB)^’ – (BA)^’ [∵ (X – Y)^’=X^’-Y^’ ] =B^’ A^’ – A^’ B^’ [∵ (AB)^’=B^’ A^’ ] =BA – AB [∵ दिया है: A^’=〖A,B〗^’=B] =- (AB-BA)
〖 ⇒(AB – BA)〗^’=-(AB – BA),
इसलिए,आव्यूह (AB – BA) एक विषम सममित आव्यूह है।

#### यदि A एक वर्ग आव्यूह इस प्रकार है कि A^2=A, तो (I+A)^3-7A बराबर है:

(I+A)^3-7A=I^3+A^3+3I^2 A+3〖IA〗^2-7A
=I+A^2 A+3IA+3IA^2-7A
[क्योंकि I^3=I^2=I] =I+AA+3A+3IA-7A
[क्योंकि A^2=A] =I+A+3A+3A-7A=I
[क्योंकि IA=A]