Matrix operations 

Matrix operations 

Matrix operations consist of three algebraic operations such as the addition of matrices, subtraction of matrices, and multiplication of matrices. A rectangular array of numbers or expressions arranged in rows and columns is known as a matrix. A wide range of applications of the matrix is found in the mathematical field. For addition or subtraction, matrices should be of the same order. The number of columns in the first matrix should be equal to the number of rows in the second matrix, for multiplication.

Matrices are used to represent real-world data. They are also used to manipulate and study linear maps between finite-dimensional vector spaces. They are also used to solve linear equations. Matrix operations are listed below. 

  • Addition 
  • Subtraction 
  • Multiplication 

Addition of Matrices

Let  A[aij]mxn and B[bij]mxn are two matrices of the same order, then their sum A + B is a matrix. Every element of that matrix is the sum of the corresponding elements. 

i.e. A + B = [aij + bij]mxn

Let us consider the two matrices A & B of order 2 x 2. Then the sum is given by:

Properties of matrix Addition:

If A, B and C are any 3 matrices of same order, then

1. Commutative property: A+B = B+A

2. Identity matrix: A + 0 = 0 + A = A. 0 is the identity matrix.

3. Associative property: A+(B+C) = (A+B)+C

4. Additive Inverse: A + -A = -A + A = 0, where -A represents the additive inverse of A.

5. If A+B = A+C, then B = C

6. Transpose of (A+B) = Transpose of A + Transpose of B

Subtraction of matrices:

Let A[aij]mxn and B[bij]mxn are two matrices of the same order then their difference A – B is a matrix. Each element of that matrix is the difference between the corresponding elements. 

A – B = A + (-B)

Let us consider the two matrices A & B of order 2 x 2. Then the difference is given by:

Scalar multiplication of matrices:

Let A = [aij] be an mxn matrix and k be any scalar, then the matrix obtained by multiplying every element of A by k is called scalar multiple of A by k and is denoted by kA.

I.e  kA= [k aij]mxn

Multiplication of matrices:

To multiply two matrices A and B, the number of columns of matrix A should be equal to the number of rows of matrix B.

If A = [aij]mxn and B = [bij]nxp are two matrices of orders m×n and n×p respectively, then their product AB is of orders m×p and is defined as

(AB)ij = ( ith row of A )( jth column of B ) for all i = 1,2,..m and j = 1,2,…p

Properties of matrix multiplication:

1. Matrix multiplication is not commutative. AB ≠ BA

2. Matrix multiplication is associative. A(BC) = (AB)C

3. Matrix multiplication is distributive. A(B+C) = AB + AC

4. The product of a matrix and a null matrix is a null matrix. 

5. There exist a multiplicative identity for every square matrix. AI = IA = A

Determinants

Determinants were developed when mathematicians were trying to solve a system of simultaneous linear equations. Determinants is a scalar value that can be calculated from the elements of a square matrix. It is an arrangement of numbers in the form

 . The horizontal lines are known as rows and vertical lines are known as columns. The shape of every determinant is a square. If a determinant is of order n then it contains n rows and n columns.

Determinant of a 3×3 matrix is determined by

 = a1(b2c3-b3c2)-b1(a2c3-a3c2)+c1(a2b3-a3b2).

The value of determinant will not change if the rows and columns are interchanged. If any two rows of a determinant are the same, then the determinant is 0. If any row of the determinant is multiplied by a variable k, then its value is multiplied by k.