Class 9 – Matrices

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Matrices
  • Introduction to matrices
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Matrices
  • Multiplication of matrices
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Introduction:

 

 

 

 

Arthur Caylay (1821 – 1895) an English mathematician created a new mathematical discipline called matrices in 1858. He used matrices to write simultaneous equations in an abridged form James Joseph Sylvester (1814 – 1897) and another English mathematician gave the name matrix (Plural – matrices) to a rectangular arrangement of numbers in rows and columns.

 

Matrix: A matrix is a rectangular array of elements. Generally the elements are numbers. (real or imaginary). The elements, are usually denoted by lower case letters a, b, c…..

Symbol of matrix: To write a matrix, we arrange the numbers in a rectangular array and enclose the array in brackets as shows below

[           ]     or  (          )

Usually capital letters of English alphabetlanguage such as A,B,….etc are used to the represent matrices.

Order of the matrix: Observe the following arrangements of real numbers.

(i)          (ii)   (iii)

In matrix (i) there are two rows and two columns. This is called a 2 by 2 matrix. This is written as 2 x 2 matrix. There are 2 rows and 3 columns in matrix (ii). It is written as 2 x 3 matrix. There may be any number be any number of rows and any number of columns in a matrix. If there are m rows and n columns in matrix,. ‘A’ it is called m x n matrix. It is written as  . This m x n of ‘A’ is called the order of the matrix ‘A’.

An m x n m x n matrix is usually written as

In compact form the above matrix is represented by  or,

The numbers a11, a12 ,…………..etc. are known as the elements of the matrix A.

The element belongs to ith  row and jth  column and is called the (i, j)th element of the matrix . Thus, in the element aij the first subscript i always denotes the number of rows and the second subscript j , number of columns in which the element occurs.

Following are some examples of matrices.

i)       is a matrix having 2 rows and 3 columns and so it is a matrix of order 2 x 3 such that

ii)      is a matrix having 2 rows and 2 columns and so it is a matrix of order 2 x 2  such that

 

Example:Construct a 3 x 4 matrix  whose elements are given by .

Solution: We have, , where

Similarly,we have

 

 

Types of matrices:

Row matrix: A matrix having only one row is called a row-matrix

For eExample: A = [1 2 -1 -2] is a row matrix of order 1 x 4.

 

 

http://images.clipartpanda.com/bloc-clipart-note-md.pngIf m x n is a order of a row matrix then m = 1 and n > 1.

 

 

Columnmatrix: A matrix having only one column is called a column matrix.

Example:  and  are column-matrices of order 3 x 1 and 4 x 1 respectively.

 

http://images.clipartpanda.com/bloc-clipart-note-md.pngIf m x n is order of a column matrix then m > 1 and n = 1

Square matrix: In a matrix, if the number of rows is equal to the numberof columns, it is called a square matrix. Its order is shown as m x m or‘m’.

Example:  is a 2 x 2 matrix. It is called a second order square matrix.

 is a 3 x 3 matrix. It is called a third-order square matrix.

A square matrix of order m isalso called a m-order square matrix. The elements  of a square matrix  for which i.e., the elements  are called the diagonal elements and the line along which they lie is called the principal diagonal or leading diagonal of the matrix.

For example, the matrix  is square matrix of order 3 in which the diagonal elements are 2, -2 and -3.

 

Diagonal matrix: A square matrix  is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero i.e.

A diagonal matrix of order  having as diagonal elements is denoted by A =diagonal

Example: The matrix  is a diagonal matrix, to be denoted by
A= diag [1 2 3].

 

Rectangular matrix: If the number of rows in a matrix is not equal to the number of columns in that matrix, it is called a Rectangular matrix.

Example: has 2 rows and 3 columns.

Here number of rows is not equal to the number of columns. Hence this is a rectangular matrix.

Example:is a rectangular matrix of order 3 x 2.

http://images.clipartpanda.com/bloc-clipart-note-md.pngA matrix is a rectangular matrix of order m x n, if and only if m x n.

 

Scalar matrix: A square matrix in which all the elements in the principal diagonal are equal and the rest of the elements are zero is called a scalar matrix.

Example:

is a scalar matrix of 3rdorder.

In general: A square matrix =  is called a scalar matrix if

i)      for all , and

ii)     for all where .

In other words, a diagonal matrix in which all the diagonal elements are equal is called the scalar matrix.

 

Example: The matrices.

 and

are scalar matrices of order  2 and 3 respectively.

 

Zero or null matrix: A matrix whose all elements are zero is called a null matrix or zero matrix. It is denoted by ‘0’.

Example:

 is a 2 x 2 null matrix. This is represented by

 is a 2 x 3 null matrix. This is represented by

Identity matrix (or) Unit matrix: A square matrix, in which each of the elements of the principal diagonal is ‘1’ and all other elements are zeroes is called an Identity matrix or Unit matrix. It is denoted by I.

Example:

 is an Identity matrix of 3rd order. It is represented by

Similarly is represented by

In general: A square matrix  is called an identity or unit matrix if

i)  for all  and,

ii)  The identity matrix of order n is denoted by In.

 

Equality of two matrices:

Definition: Two matrices are equal if and only if they are of the same order and their corresponding elements are equal.

Example:

State whether the following matrices are equal or not.

i)     

Solution:

i)      Matrices A and B are of the same order. But their corresponding elements are different. Hence .

ii)     Matrices C and D are of the same order and their corresponding elements are also equal C = D.

In general: Two matrices  and  are said to be equal if

i)     m = r, i.e., the number of rows in A is equal to the number of rows in B.

ii)      i.e., the number of columns in A is equal to the number of columns in B.

iii)    for  and .

http://images.clipartpanda.com/bloc-clipart-note-md.pngIf two matrices A and B are equal, we write A = B, otherwise we write .

 

Example:If  find x, y, z,w.

Solution: Since the corresponding elements of two equal matrices are equal. Therefore,

Solving the equations  and  as simultaneous linear equations,
we get x = 1, y = 2.

Now,putting x = 1 in 2. x + z = 5, we get z = 3.

Substituting z = 3 in 3. z + w = 13, we obtain w = 4.

Addition of matrices: Let A, B be two matrices, each of order Then their sum
A + B is a matrix of order  and is obtained by adding the corresponding elements of A and B.

Thus, if  and  are two matrices of the same order, their sum
A + B is defined to be the matrix of order  such that

 for

http://images.clipartpanda.com/bloc-clipart-note-md.pngThe sum of two matrices is defined only when they are of the same order.

 

Example: If , then

Example: If , then A + B is not defined, because A and B are not of the same order.

Properties of matrix addition:

1.    Matrix addition is closure: If A and B are two matrices of order m x n then their sum (A + B) is also a matrix.

2.    Matrix addition is commutative:i.e., if A and B are two matrices of order m x n, then A + B = B + A.

Proof: Let  be two matrices of order m x n. Then, A + B and B + A both are m x n  matrices such that

For

Thus, A + B and B + A are two matrices such that their orders are same and the corresponding elements are equal.

Hence,A + B = B + A

http://images.clipartpanda.com/bloc-clipart-note-md.pngTo prove that two matrices are equal it is required to prove that their orders are same and the corresponding elements areequal.

3.    Matrix addition is associative: If A, B, C are three matrices of the same order, then

.

Proof: Let , B =  and C =  be three m x n matrices.

Then, (A + B) + C and A + (B + C) are m x n matrices such that

  =

                    =

                    =  =

                    =  for

Thus, (A + B) + C and A + (B+ C) are two matrices such that their orders are same and the corresponding elements are equal.

Hence,(A + B) + C = A + (B + C)

4.    Existence of Identity: The null matrix is the identity matrix for matrix addition, i.e., A + O = A = O +A.

Proof: Let  be any matrix of order  and O be a null matrix of order . Then, A + O and O +A are matrices of order  such that

and,

Hence,

5.    Existence of Inverse: For every matrix  there exists a matrix ,denoted by, such that .

Proof:We have,

and,for all

Hence,

The matrix  is called the additive inverse of the matrix  and vice-versa.

6.    Cancellation property: Cancellation laws hold good in case of addition of matrices. If A, B, C are matrices of the same order, then  (left cancellation law) and  (right cancellation law)

Proof:

 

Multiplication of a matrix by a scalar (scalar product of a matrix): Let  be a  matrix of order  and k be any real number calleda scalar. Then the matrix obtained by multiplying every element of A by k is called the scalar multiple of A by k and is denoted by  Thus,

Example: If , then

 

Properties of scalar multiplication: If  are two matrices and
k, l are scalars, then

i)                     ii)

iii)                iv)

v)                                    vi)

Subtraction of matrices: For two matrices A and B of the same order, we define
.

Example: If   and  then

 

Example: If , , find 3A – 2B

Solution:We have,

3A– 2B = 3A + (-2) B

        

Example: Find X and Y, if  and

Solution: We have,  and

And,

 

 

Transpose of matrix: The transpose of a given matrix is a matrix abstained by interchanging the rows and columns. In general.  Let  be a matrix of order . Then the transpose of A, denoted by AT or A1, is a matrix of order such that

 for all

Thus,is obtained from A by changing its rows into columns and its columns into rows.

For example, if , then

http://images.clipartpanda.com/bloc-clipart-note-md.pngThe first row of  AT is the first column of A. The second row of AT is the second column of A and so on.

Properties of transpose: Let A and B be two matrices. Then,

i)

ii), A and B being of the same order.

iii) be any scalar(real or complex)

iv), A and B being conformable for the product AB.

Special types of square matrices:

Symmetric matrix: If a matrix is equal to its transpose then the matrix is called symmetric matrix. A square matrix  is called symmetric matrix if

.

Example: The matrix  is symmetric because

It follows from the definition of a symmetric matrix that A is symmetric.

Thus,a square matrix A is a symmetric matrix iff AT = A.

Example: , B =  are symmetric matrices because  and.

Skew-symmetric matrix: If the transpose of a matrix is equal to the additive inverse of the given matrix, then the given matrix is called skew symmetric matrix.

Ingeneral: A square matrix  is a skew symmetric matrix if

.

Example: The matrix  is skew-symmetric, because

It follows from the definition of a skew-symmetric matrix that A is skew-symmetric.

Thus,a square matrix A is a skew-symmetric matrix if .

Example:,  are skew symmetric matrices because  and.

 

http://www.aids.harvard.edu/img/news/spotlight/newsletter_layout/logo.gif

Example: Let A be a square matrix. Then
i)  is a symmetric matrix                    ii) is askew-symmetric matrix
iii)  are symmetric matrices.
Solution:i) Let .Then

ii) . Then

iii) We have