Matrix refer to a rectangular formation (array or table) of certain number(real or complex) or of certain function.
Matrics are represented bt the capital letters A, B, C, etc.
Order of matrix
A matrix having m rows and n columns and enclosed by a square bracket [ ] is called mxn matrix (read “m by n matrix”).
An mxn matrix is expressed as
or A = (aij)m x n
For Example: The order or dimension of a matrix is the ordered pair having as a first component the number of rows and as a second component the number of columns in the matrix. If there are 3 rows and 2 columns in a matrix, then its order is written as (3, 2) or (3 x 2) read as three by two. In general, if m is rows and n are columns of a matrix, then its order is (m x n).
The m,n numbers are known as Element of the matrix. They are accompanied by two indices the first of which indicated the row on which it appears and second the appropriate column.
are matrices of orders (2 x 3), (3 x 1) and (4 x 4) respectively.
Types of Matrices
A matrix of order 1 x n is called a row matrix or a row vector. Its consisting of a single row is called a row matrix or a row vector.
A = [1, 2, 3]
A matrix of order m x 1 is called a column matrix or a column vector. Its consisting of a single column.
Null or Zero Matrix:
A matrix in which each element is “0” is called a Null or Zero matrix. Zero matrices are generally denoted by the symbol O. This distinguishes zero matrix from the real number 0.
is a zero matrix of order 2 x 4.
The matrix Om x n has the property that for every matrix Am x n,
A + O = O + A = A
A matrix having same numbers of rows and columns is called a square matrix. A matrix A of order m x n can be written as Am x n. If m = n, then the matrix is said to be a square matrix. A square matrix of order n x n, is simply written as An.
are square matrix of order 2 and 3
Main or Principal (leading)Diagonal
The principal diagonal of a square matrix is the ordered set of elements aij, where i = j, extending from the upper left-hand corner to the lower right-hand corner of the matrix. Thus, the principal diagonal contains elements a11, a22, a33 etc.
For example, the principal diagonal of
consists of elements 1, 2 and 0, in that order.
Cases of a square matrix:
A square matrix in which all elements are zero except those in the main or principal diagonal is called a diagonal matrix. Some elements of the principal diagonal may be zero but not all.
are diagonal matrices.
is a diagonal matrix if and only if
aij = 0 for i ≠ j
aij ≠ 0 for at least one i = j
A diagonal matrix in which all the diagonal elements are same, is called a scalar matrix.
Identity Matrix or Unit matrix
A scalar matrix in which each diagonal element is 1(unity) is called a unit matrix. An identity matrix of order n is denoted by In.
are the identity matrices of order 2 and 3.
is an identity matrix if and only if
aij = 0 for i ≠ j and aij = 1 for i = j
Note: If a matrix A and identity matrix I are comformable for multiplication, then I has the property that AI = IA = A i.e., I is the identity matrix for multiplication.
Two matrices A and B are said to be equal if and only if they have the same order and each element of matrix A is equal to the corresponding element of matrix B i.e for each i, j, aij = bij
then A = B because the order of matrices A and B is same and aij = bij for every i, j.
Upper Triangular Matrix
If in a matrix, elements for which i > j are zeros, then it is known as upper triangular matrix.
Lower Triangular Matrix
If in a matrix, elements for which i < j are zeros, then it is known as lower triangular matrix.
The Negative of a Matrix:
The negative of the matrix Am x n, denoted by –Am x n, is the matrix formed by replacing each element in the matrix Am x n with its additive inverse.
for every matrix Am x n, the matrix –Am x n has the property that
A + (–A) = (–A) + A = 0
i.e., (–A) is the additive inverse of A.
The sum Bm – n + (–Am x n) is called the difference of Bmxn and Amxn
and is denoted by Bm x n – Am x n.
Operations on matrices
Addition of Matrix:
Two matrices can be added only if they are of the same order. In such case, the respective elements of the two matrices are added e.g
if A = |aij|m x n , B = |bij|m x n
then A + B = |aij + bij|m x n
Properties of Matrix addtion
• It is commutative i.e. A + B = B + A
• It is associative i.e. ( A + B ) + C = A + ( B + C )
• It is odditive i.e. A + 0 = 0 + A = A
• Existance of additive inverse i.e. A + (-A) = 0 = (-A) + A
Subtraction of Matrix:
Two matrices can be subtract only if they are of the same order. In such case, the respective elements of the two matrices are subtracted e.g
if A = |aij|m x n , B = |bij|m x n
then A – B = |aij – bij|m x n
Multiplication of Matrix:
Matrix multiplication is possible only if the number of columns of the first matrix is equal to the number of rows of the second matrix.
Properties of Matrix Multiplication
• It is associative i.e. ( AB )C = A( BC )
• It is distributive i.e. A( B + C ) = AB + AC
or (A + B)C = AC +BC
1 . Multiplication of matrices is not commutative i.e., AB ≠ BA in general.
2 . For matrices A and B if AB = BA then A and B commute to each other
3 . A matrix A can be multiplied by itself if and only if it is a square matrix.The product A.A in such cases is written as A2. Similarly we may define higher powers of a square matrix i.e.,
A . A2 = A3 , A2. A2 = A4
4. In the product AB, A is said to be pre multiple of B and B is said to be post multiple of A.
Multiplication of a scalar to a matrix
If A = (aij)m x n and k is a scalar, then
k.A = (k aij)m x n
If we multiply a matrix by some scalr, the all the element of the matrix are multiplied by scalar.
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