Matrix operations mainly involve three algebraic operations which are
addition of matrices, subtraction of matrices, and multiplication of
matrices. Matrix is a rectangular array of numbers or expressions
arranged in rows and columns.
As far as linear algebra is concerned, the two most important operations with vectors are vector addition [adding two (or more) vectors] and scalar multiplication (multiplying a vector by a scalar).
Operations
Addition, subtraction and multiplication are the basic operations on the matrix. To add or subtract matrices, these must be of identical order and for multiplication, the number of columns in the first matrix equals the number of rows in the second matrix.
- Addition of Matrices
- Subtraction of Matrices
- Scalar Multiplication of Matrices
- Multiplication of Matrices
Matrix Addition
If A[aij]mxn and B[bij]mxn are two matrices of the same order then their sum A + B is a matrix, and each element of that matrix is the sum of the corresponding elements.
i.e. A + B = [aij + bij]mxn
For the Example:
Properties of Matrix Addition: If a, B and C are matrices of same order, then
(a) Commutative Law: A + B = B + A
(b) Associative Law: (A + B) + C = A + (B + C)
(c) Identity of the Matrix: A + O = O + A = A, where O is zero matrix which is additive identity of the matrix,
(d) Additive Inverse: A + (-A) = 0 = (-A) + A, where (-A) is obtained by changing the sign of every element of A which is additive inverse of the matrix,
(e)
(f)
(g) If A + B = 0 = B + A, then B is called additive inverse of A and also A is called the additive inverse of A.
Subtraction of Matrices
If A and B are two matrices of the same order, then we define
Scalar Multiplication of Matrices
If is a matrix and k any number, then the matrix which is obtained by multiplying the elements of A by k is called the scalar multiplication of A by k and it is denoted by k A thus if
Then
Properties of Scalar Multiplication: If A, B are matrices of the same order and λ and μ are any two scalars then;
(a)
(b)
(c)
(d)
(e)
Multiplication of Matrices
If A and B be any two matrices, then their product AB will be defined only when the number of columns in A is equal to the number of rows in B.
The first step in defining matrix multiplication is to recall the definition of the dot product of two vectors. Let r and c be two n‐vectors. Writing r as a 1 x n row matrix and c as an n x 1 column matrix, the dot product of r and c is
Properties of matrix multiplication
(a) Matrix multiplication is not commutative in general, i.e. in general
(b) Matrix multiplication is associative, i.e. (AB)C = A(BC).
(c) Matrix multiplication is distributive over matrix addition, i.e. A.(B + C) = A.B + A.C and (A + B)C = AC + BC.
(d) If A is an m × n matrix, then
(e) The product of two matrices can be a null matrix while neither of them is null, i.e. if AB = 0, it is not necessary that either A = 0 or B = 0.
(f) If A is an m × n matrix and O is a null matrix then i.e. the product of the matrix with a null matrix is always a null matrix.
(g) If AB = 0 (It does not mean that A = 0 or B = 0, again the product of two non-zero matrices may be a zero matrix).
(h) If AB = AC , B ≠ C (Cancellation Law is not applicable).
(i)
j) There exist a multiplicative identity for every square matrix such AI = IA = A
Matrix Transpose
If A is m × n, the transpose of A is the n × m matrix, denoted by A^T , whose columns are formed from the corresponding rows of A.
A square matrix has the same number of rows and columns. An identity matrix is a square matrix with ones on the diagonal from upper left to lower right and zeros elsewhere. For example:
I =
1 0 0
0 1 0
0 0 1
Such a matrix is often denoted I.
REF
https://byjus.com/jee/matrix-operations/
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