Online Operations on Matrices Assignment Help & Homework Help
Matrix Addition, Multiplication Tutoring Online, Tutor Help
OPERATIONS ON MATRICES
1) MATRIX ADDITION:
The sum of two matrices is a matrix obtained by adding the corresponding elements of the given matrix.
If
is a 2 x 2 matrix and
is another 2 x 2 matrix, then
Or
If A = [aij] and B = [bij] then,
A + B = [aij] + [bij] = [aij + bij] for all i and j.
Example: Given
and 
, find A + B.
Solution:
If
is a 2 x 2 matrix and
is another 2 x 2 matrix, then
Or
If A = [aij] and B = [bij] then,
A + B = [aij] + [bij] = [aij + bij] for all i and j.
Example: Given
and , find A + B.Solution:
PROPERTIES OF MATRIX ADDITION:
- Commutative Property: If A = [aij] and B = [bij] are two matrices of same order, then A + B = B + A.
- Associative Property: If A = [aij] and B = [bij] and C = [cij] are matrices of same order, then (A + B) + C = A + (B + C).
- Additive Identity: If A = [aij] be a matrix of order mx n and O be a zero matrix of order mx n, then A + O = O + A = A. Hence the matrix O is the additive identity for matrix addition.
- Additive Inverse: For every matrix A = [aij]mxn , there exists a matrix -A = [-aij]mxn such that A + (-A) = 0. The matrix -A is known as the additive inverse of A.
2) MATRIX MULTIPLICATION:
MULTIPLICATION OF A MATRIX BY A SCALAR:
If A = [aij]mxn is a matrix and k is a scalar, then k.A is another
matrix which is obtained by multiplying each element of A by the scalar k, that is,
k.A = k[aij]mxn = [k(aij)]mxn which means that (i, j)th element of kA is kaij.
Example:
If
then find 2A.
Solution:
k.A = k[aij]mxn = [k(aij)]mxn which means that (i, j)th element of kA is kaij.
Example:
If
then find 2A.Solution:
MULTIPLICATION OF MATRICES:
Two matrices A and B can be multiplied only if the number of columns of A is equal
to the number of rows of B. Hence, if A is a m x n matrix, then to find the product of
A with a matrix B, the matrix B must have n rows and any finite number of columns.
Let A be a matrix of order m x n and B is a matrix of order n x p, then the product matrix C of matrices A and B will be:
A x B = C
Hence the order of the product matrix C will be m x p.
To find the (i, j)th element of C which is cij, the ith row of A and jth column of B are taken and multiplied (element wise) and then the sum of all these products is obtained.
Hence, ith row of A = [ai1 ai2 ai3 .... ain]
And the jth column of
Hence cij = ai1. b1j + ai2. b2j + ai3. b3j + .... + ain. bnj
=
Example:
If
and 
, then find AB.
Solution:
Hence
PROPERTIES OF MULTIPLICATION OF MATRICES:
3) ELEMENTARY OPERATIONS ON A MATRIX:
There are 3 operations which can be performed on matrices. Three of these operations are due to rows and three are due to columns. These are known as elementary operations or transformations.
Let A be a matrix of order m x n and B is a matrix of order n x p, then the product matrix C of matrices A and B will be:
A x B = C
Hence the order of the product matrix C will be m x p.
To find the (i, j)th element of C which is cij, the ith row of A and jth column of B are taken and multiplied (element wise) and then the sum of all these products is obtained.
Hence, ith row of A = [ai1 ai2 ai3 .... ain]
And the jth column of
Hence cij = ai1. b1j + ai2. b2j + ai3. b3j + .... + ain. bnj
=
Example:
If
and , then find AB.Solution:
Hence
PROPERTIES OF MULTIPLICATION OF MATRICES:
- Associative Property: If A, B and C are any three compatible matrices, then A(BC) = (AB)C.
- Distributive Property: If A, B and C are any three compatible matrices, then
- A(B + C) = AB + BC
- (A + B)C = AC + BC
- Multiplicative Identity: For every square matrix A, there exists an identity matrix of the same order such that IA = AI =A.
3) ELEMENTARY OPERATIONS ON A MATRIX:
There are 3 operations which can be performed on matrices. Three of these operations are due to rows and three are due to columns. These are known as elementary operations or transformations.
- Interchange of any two rows or columns: If ith row (column) of a matrix is interchanged with the jth row (column), then it is represented by Ri ↔ Rj (Ci ↔ Cj).
- The multiplication of the elements of any row or column by a non-zero scalar: If the elements of ith row (column) are multiplied by a non-zero scalar, then it is represented byRi → k.Ri (Ci → k.Ci).
- The addition to the element of any row (column), the corresponding elements of any other row or column multiplied by any non-zero scalar. If k times the elements of jth row (column) are added to the corresponding elements of the ith row (column), then it is denoted by
Ri → Ri + k.Rj (Ci → Ci+ k.Ci).
Online Operations on Matrices Help:
If you are stuck with a Operations on Matrices Homework problem and need help, we have excellent
tutors who can provide you with Homework Help. Our tutors who provide Operations on Matrices help
are highly qualified. Our tutors have many years of industry experience and have had years of
experience providing Operations on Matrices Homework Help. Please do send us the Operations on
Matrices problems on which you need Help and we will forward then to our tutors for review.
Online Tutor Operations on Matrices:
We have the best tutors in math in the industry. Our tutors can break down a complex Operations on
Matrices problem into its sub parts and explain to you in detail how each step is performed. This
approach of breaking down a problem has been appreciated by majority of our students for learning
Operations on Matrices concepts. You will get one-to-one personalized attention through our online
tutoring which will make learning fun and easy. Our tutors are highly qualified and hold advanced
degrees. Please do send us a request for Operations on Matrices tutoring and experience the quality
yourself.
