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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.
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
\[\begin{bmatrix} a_{11} & a_{12} & a_{13} &...... & a_{1n} \\ a_{21} & a_{22} & a_{23} &...... & a_{2n} \\ a_{31} &a_{32} & a_{33} &...... & a_{3n} \\ ....& .... & ... & ...... & .... \\ ....& .... & ... & ...... & .... \\ a_{m1}&a_{m2} &a_{m3} & ...... &a_{mn} \end{bmatrix}_{m\times n}\]
a11
means an element in the first row and first column, a23 means an
element in the second row and in the third column and so on.
In short a
matrix can be written as A = [aij]m
x n , where aij 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'
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.
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
⇒ 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.
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 Ri ↔ Rj
ii) Multiplication of
all the members of any row by a scalar k is is denoted by Ri → kRi
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 Ri → Ri + kRj
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 a11 = 1
Step 3: With the help of a11 = 1 make a21 =
0
Step 4: Now make a22 = 1
Step 5: With the help of a22 = 1 make a12 =
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 a11 = 1
Step 3: With the help of a11 = 1 make a21 and
a31 = 0
Step 4: Now make a22 = 1
Step 5: With the help of a22 = 1 make a12 and
a32 = 0
Step 6: Now make a33 = 1
Step 7: With the help of a33 = 1 make a13 and
a23 = 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
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-1A-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-1A) B(AB)-1 = A-1
(IB)(AB)-1 = A-1 ⇒ B(AB)-1 = A-1
(B-1B) (AB)-1
= B-1A-1 ⇒ I(AB)-1 = B-1A-1
⇒ (AB)-1 = B-1A-1
⇒ (AB)-1 = B-1A-1
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