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### Chapter 3 Matrix Class XII : Basic Concepts

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__Class 12 Chapter 3 Matrices__

*Complete explanation, basic concepts and formulas based on matrices. Matrix method of arranging the terms , addition, subtraction, multiplication , and transpose of the matrices.*

**Solution of Important questions of NCERT Chapter 3 Matrices**

__Basic Concepts Based on Chapter 3 (Matrix) Class 12__

**Definition :**

**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.**

**Order of a Matrix:**

**A matrix having m rows and n columns is called a matrix of order m x n**

**In general a matrix of order m x n can be arranged in a rectangular array as**

**\[\begin{bmatrix} a_{11} & a_{12} & a_{13} &...... & a_{1n} \\ a_{21} & a_{22} & a_{23} &...... & a_{2n} \\ a_{31} &a_{32} & a_{33} &...... & a_{3n} \\ ....& .... & ... & ...... & .... \\ ....& .... & ... & ...... & .... \\ a_{m1}&a_{m2} &a_{m3} & ...... &a_{mn} \end{bmatrix}_{m\times n}\]**

**a**

_{11}means an element in the first row and first column, a_{23}means an element in the second row and in the third column and so on.

**In short a matrix can be written as A = [a**

_{ij}]_{m x n}, where a_{ij}represent the elements of the matrix, m is the number rows and n is the number of columns.

__Types of Matrices__

**Column Matrix :**

**If the matrix has only one column then it is called column matrix.**

**\[\begin{bmatrix} a\\b \\c \end{bmatrix}\]**

**Row Matrix : If a matrix has only one row then it is called a row matrix.**

**\[\begin{bmatrix} a & b &c \end{bmatrix}\]**

**Square Matrix :**

**A matrix in which number of rows and columns are equal is called a square matrix. For Example:**

**\[\begin{bmatrix} a &b \\c &d \end{bmatrix}_{2\times 2}\; \; \; or\; \; \; \begin{bmatrix} a &b &c \\d & e &f \\ 1& 2 &3 \end{bmatrix}_{3\times 3}\]**

**Diagonal Matrix :**

**A square matrix is said to be diagonal matrix if its all non diagonal elements are zero. For example :**

**\[\begin{bmatrix} a &0 &0 \\ 0 &b &0 \\ 0 & 0 & c \end{bmatrix}\]**

**Scalar Matrix :**

**A diagonal matrix is said to be scalar matrix if its all diagonal elements are equal. For Example :**

\[\begin{bmatrix} 5 &0 &0 \\ 0 &5 &0 \\ 0 & 0 & 5 \end{bmatrix}\]

**Identity Matrix :**

**A diagonal matrix is said to be identity matrix if its all diagonal elements are 1. Example :**

**\[\begin{bmatrix} 1 &0 &0 \\ 0 &1 &0 \\ 0 & 0 & 1 \end{bmatrix}\]**

**Zero Matrix :**

**A square matrix is said to be zero matrix if its all elements are zero.**

**\[O=\begin{bmatrix} 0 &0 &0 \\ 0 &0 &0 \\ 0 & 0 & 0 \end{bmatrix}\]**

**Equal Matrices :**

**Two matrices A and B are said to be equal if**

**a) Matrices A and B are of same order**

**b) Corresponding elements of matrix A and B are equal.**

__Addition of two matrices :__

**Addition of two matrices is possible if they are of same order**

**Addition of two matrices is done by adding their corresponding elements.**

**Example : \[\begin{bmatrix} a &b &c \\ x& y & z\\ p & q & r \end{bmatrix}+\begin{bmatrix} 1&2 &3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}=\begin{bmatrix} a+1&b+2 &c+3 \\ x+4 & y+5 & z+6\\ p+7 & q+8 & r+9 \end{bmatrix}\]**

**Addition of two matrices is not possible if they are not of same order.**

__Multiplication of a matrix with a scalar__

**Let A is a matrix and k is a scalar, then multiplication of scalar k with matrix A is obtained by multiplying each element of matrix A by a scalar k.**

**\[If\; matrix\; A=\begin{bmatrix} x &y &z \\ p & q &r \\ a&b &c \end{bmatrix} \; and \; scalar\; k=2,\; then\] \[2A=2\begin{bmatrix} x &y &z \\ p & q &r \\ a&b &c \end{bmatrix}=\begin{bmatrix} 2x &2y &2z \\ 2p & 2q &2r \\ 2a & 2b &2c \end{bmatrix}\]**

__Properties of matrix addition__

**If A and B are two matrices then**

**Commutativity**

**Matrix addition holds the commutative law : A + B = B + A**

**Associative law**

**Matrix addition holds the associative law : (A + B) + C = A + (B + C)**

**Existence of Identity : If zero matrix O is added to any matrix A then matrix A is remain unchanged. A + O = O + A = A ⇒ O is the identity matrix in Matrix addition.**

**Existence of additive inverse:**

**If A is a matrix then A + (-A) = (-A) + A = O, then -A is said to be the Additive inverse of matrix A.**

__Multiplication of matrices__

**Product of two matrices A and B is defined if number of columns of first matrix A is equal to the number of rows of matrix B**

**Example :**

**If A and B are two matrices such that:**

\[A=\left [ a_{ij} \right ]_{m\times n}\; \; and\; \; B=\left [ b_{ij} \right ]_{p\times q}\]

**Then product AB exists if n = p and the order of the resulting matrix AB is m x q**

**Note :**

**If product AB is defined then product BA need not to be defined**

**Product AB and BA both are defined if A and B both are the square matrices of same order.**

**Example : If A and B are two matrices such that :**

**\[A = \begin{bmatrix} 2 &4 &-3 \\ 1 &4 & 3 \end{bmatrix}\: \: and\: \: B=\begin{bmatrix} 2 &-3 \\ 4 &5 \\ 1 & 2 \end{bmatrix}\]**

**Order of matrix A is 2 x 3, and the order of matrix B = 3 x 2.**

**Number of columns of first matrix is equal to the number of rows of second matrix.**

**Therefore product of A and B is defined. And the order of the resulting matrix become 2 x 2**

\[AB=\begin{bmatrix} 2\times 2+4\times4+(-3)\times1 &\; \; \; \; \; 2\times-3+4\times5+(-3)\times2 \\ 1\times2+4\times4+3\times1&1\times-3+4\times5+3\times2 \end{bmatrix}\]

\[AB=\begin{bmatrix} 4+16-3 &\; \; -6+20-6 \\ 2+16+3 &\; \; -3+20+6 \end{bmatrix}\]

**\[AB=\begin{bmatrix} 17 &\; \; 8 \\ 21 &\; \; 23 \end{bmatrix}\]**

__Properties of matrix multiplication.__

**i) Matrix multiplication is not commutative.**

**⇒ AB ≠ BA**

**ii) If A and B are the diagonal matrices of the same order then multiplication of the matrices is commutative. In this case we can write AB = BA**

**iii) Matrix multiplication is associative**

**⇒**

**(AB)C = A(BC)**

**iv) Matrix multiplication holds the Distributive property**

**A(B + C) = AB + AC or (A+B)C = AC + BC**

**Existence of multiplicative identity :**

**For every square matrix there exists an identity matrix I of the same order such that AI = IA = A,**

**⇒**

**I is the identity matrix for A.**

**Example**

**For a square matrix of 2, I is the identity matrix and is given by \[I=\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}=I_{3}\]**

**For a square matrix of 3, I is the identity matrix and is given by**

**\[I=\begin{bmatrix} 1 &0 &0 \\ 0&1 &0 \\ 0& 0 &1 \end{bmatrix}=I_{3}\]**

__Transpose of a matrix__

**If A is a matrix then transpose of the matrix A is obtained by interchanging the rows and columns . It is denoted by A'**

**Example**

\[If\; A=\begin{bmatrix} 17 & 8\\ 21& 23 \end{bmatrix}\; \; then\; \; A' =\begin{bmatrix} 17 &21 \\ 8& 23 \end{bmatrix}\]

\[If\; \; A=\begin{bmatrix} a &b &c \\ x& y & z\\ p &q &r \end{bmatrix}\; \; then,\; A'=\begin{bmatrix} a & x &p \\ b& y &q \\ c& z &r \end{bmatrix}\]

__Properties of the transpose of the matrices__

**i) (A')' = A, or Transpose of the transpose matrix is the matrix itself**

**ii) (kA)' = kA', where k is any scalar**

**iii) (A + B)' = A' + B'**

**Orthogonal Matrices**

Let A be a matrix and A' is its transpose.

A square matrix of order n is said to be orthogonal if its transpose is equal to its inverse.

or

A square matrix of order n is said to be orthogonal if product of its transpose and its inverse gives a unit matrix

Mathematically: A'A-1 = I

**Symmetric and Skew symmetric matrices**

**Symmetric Matrix :**

**A matrix A is said to be symmetric if A' = A**

**Skew symmetric matrix :**

**A matrix is said to be skew symmetric if A' = -A**

**Theorem 1:**

**For a square matrix A with real entries A + A' is always symmetric matrix and A - A' is always skew symmetric matrix.**

**Proof:**

**Let A + A' = B, then**

**B' = (A + A')' = A' + (A')' = A' + A = A + A' = B**

**⇒ B = A + A' is a symmetric matrix**

**Let C = A - A' , then**

**C' = (A - A')' = A' - (A')' = A' - A = - (A - A') = - C**

**Theorem 2:**

**Any square matrix can be expressed as the sum of symmetric and skew symmetric matrix.**

**Proof:**

**If A is a matrix and A' is a transpose of matrix A, then A + A' is always a symmetric matrix and A - A' is always a skew symmetric matrix**

**\[\Rightarrow P=\frac{1}{2}(A+A')\; is\; a\; symmetric\; matrix\]**

**\[\Rightarrow Q=\frac{1}{2}(A-A')\; is\; a\; skew\; \; symmetric\; matrix\]**

**\[A=P+Q=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')=A\]**

**⇒ Every square matrix can be written as the sum of symmetric and skew symmetric matrix**

**Elementary operations of a matrix**

**There are mainly six operations on the matrix. Three of which are due to row and other three of which are due to column.**

**Here we explain only row operations. Column operations are also same.**

**i) Interchanging any two rows is denoted by**

**R**

_{i}**↔**

**R**

_{j}

_{}**ii) Multiplication of all the members of any row by a scalar k is is denoted by**

**R**

_{i}→ kR_{i}

_{}**iii) The addition of the elements of any row with the corresponding elements of any other row multiplied by any non-zero number is denoted by**

**R**

_{i}**→**

**R**

_{i}+ kR_{j}

**To find the inverse of the matrix by the method of elementary operations**

**To find A**

^{-1}by elementary operations we can either follow row operations or column operations.

**If we follow row operations then we write the equation A = IA**

**If
we follow column operations then we write the equation A = AI**

**Method of Finding the Inverse of 2 x 2 matrix
by Elementary Operations**

**Step 1: For Row operations we write A = IA**

**Step 2: Make a _{11} = 1**

**Step 3: With the help of a _{11} = 1 make a_{21} =
0**

**Step 4: Now make a _{22} = 1**

**Step 5: With the help of a _{22} = 1 make a_{12} =
0**

**Method of Finding the Inverse of 3 x 3 matrix
by Elementary Operations**

**Step 1: For Row operations we write A = IA**

**Step 2: Make a _{11} = 1**

**Step 3: With the help of a _{11} = 1 make a_{21 }and
a_{31} = 0**

**Step 4: Now make a _{22} = 1**

**Step 5: With the help of a _{22} = 1 make a_{12} and
a_{32} = 0**

**Step 6: Now make a _{33} = 1**

**Step 7: With the help of a _{33} = 1 make a_{13} and
a_{23} = 0**

__Invertible
matrices__

**If A and B are two square matrices of the same order such that AB =BA = I, then B is said to be inverse of A and is denoted by A**

^{-1}such that A^{-1 }= B. In this case A is said to be invertible.

**In other words :**

**If A is invertible then A**

^{-1}exists or If A^{-1}exists then A is said to be invertible.

**Theorem 3 :**

**Inverse of a matrix if exist then it is unique.**

**Proof:**

**If possible let us suppose that the matrix A has two inverse B and C**

**B is the inverse of A then AB = BA = I………… (i)**

**C is the inverse of A then AC = CA = I ………(ii)**

**From (i) and (ii) we get**

**AB = AC ⇒**

**B = C ………….(Cancelling A on both side)**

**⇒**

**Both the inverse are same.**

**Theorem 4:**

**If A and B are two invertible matrices of the same order then (AB)**

^{-1}= B^{-1}A^{-1}

**Proof :**

**By the definition of inverse we have**

**(AB)(AB)**

^{-1 }= I**Multiplying on both side by A**

^{-1 }we get

**A**

^{-1}(AB) (AB)^{-1}= A^{-1 }I**⇒**

**(A**

^{-1}A) B(AB)^{-1}= A^{-1}**(IB)(AB)**

^{-1}= A^{-1}**⇒**

**B(AB)**

^{-1}= A^{-1}**(B**

^{-1}B) (AB)^{-1 }= B^{-1}A^{-1}**⇒**

**I(AB)**

^{-1 }= B^{-1}A^{-1}

**⇒**

**(AB)**

^{-1 }= B^{-1}A^{-1}

^{}

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