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Operations On Matrices

Operations on Matrices Online Tutoring / Homework Help
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:



PROPERTIES OF MATRIX ADDITION:
  1. Commutative Property: If A = [aij] and B = [bij] are two matrices of same order, then A + B = B + A.


  2. Associative Property: If A = [aij] and B = [bij] and C = [cij] are matrices of same order, then (A + B) + C = A + (B + C).


  3. 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.


  4. 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:



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:

  1. Associative Property: If A, B and C are any three compatible matrices, then A(BC) = (AB)C.

  2. Distributive Property: If A, B and C are any three compatible matrices, then

    1. A(B + C) = AB + BC

    2. (A + B)C = AC + BC

  3. 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.

  1. Interchange of any two rows or columns:
  2. If ith row (column) of a matrix is interchanged with the jth row (column), then it is represented by Ri ↔ Rj (Ci ↔ Cj).

  3. The multiplication of the elements of any row or column by a non-zero scalar:
  4. If the elements of ith row (column) are multiplied by a non-zero scalar, then it is represented byRi → k.Ri (Ci → k.Ci).

  5. The addition to the element of any row (column), the corresponding elements of any other row or column multiplied by any non-zero scalar.
  6. 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).


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