Class 9 – 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 rowmatrix
For eExample: A = [1 2 1 2] is a row matrix of order 1 x 4.
If 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 columnmatrices of order 3 x 1 and 4 x 1 respectively.
If 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 thirdorder square matrix.
A square matrix of order m isalso called a morder 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.
A 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 .
If 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
The 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
To 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 viceversa.
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 A^{T} or A^{1}, 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
The 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.
Skewsymmetric 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 skewsymmetric, because
It follows from the definition of a skewsymmetric matrix that A is skewsymmetric.
Thus,a square matrix A is a skewsymmetric matrix if .
Example:, are skew symmetric matrices because and.
Example: Let A be a square matrix. Then
i) is a symmetric matrix ii) is askewsymmetric matrix
iii) are symmetric matrices.
Solution:i) Let .Then
ii) . Then
iii) We have