Matrices in Math

Introduction to matrices in math:

In order to locate the position of a particular element of a matrix. We have to specify  the number of the row and that of  the column in which the element occurs. An element occurring in the ith row and jth column of a matrix A will be called the (i,j)th element of A , to be denoted by aij .

In general ,an m x n matrix A may be written as ,

A     =     `[[a_11,a_12,...,a_1n],[a_21,a_22,...,a_2n],[...,...,...,...],[a_m1,a_m2,...,a_mn]]`

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Types of matrices:


Row matrix:

Row matrix is a matrix that contain only one row or a row vector.

Example :

(i) A = [aij]1 × 3 = [1 6 5] is a row matrix of order 1 × 3.

(ii) B = [bij]1 × 2 = [0  1] is a row matrix of order 1 × 2

(iii) C = [cij]1 × 1 = [1] is a row matrix of order 1 × 1

Column matrix:

A colum matrix is a matrix that contain  only one column  or a column vector.

Example :

1)[aij]mxn =`[[2],[4],[6]]` is a column matrices of order 3 x1

2)[aij]mxn =`[[0],[1]]` is a column matrices of order 2 x1

3)[aij]mxn =[45] is column matrices of order 1 x1

Square matrix:

A square  matrix is a matrix in which the elments in rows and columns are equal and a  matrix of order n × n is also known as a square matrix of order n.

Example :

1) `[[4,5],[5,4]]` is the square matces of order 2 x2

2) `[[1,2,3],[4,5,6],[7,8,9]]` is the square matrix of order 3 x3

3) `[[0,1,2,3],[2,1,4,5],[3,4,6,7],[0,6,7,8]]` is the square matrix of order 4 x4.

Diagonal matrix:

Diagonal matrix means  a  square matrix A = [aij]n × n is said to be a diagonal matrix if aij = 0 when i ? j. In a diagonal matrix all the entries except the entries along the main diagonal are zero.

Example :

1) `[[1,0,0],[0,2,0],[0,0,3]]` is the diagonal matrix of order 3 x3

2) `[[3,0],[0,7]]`   is the diagonal matrix of order 2 x2.

Zero matrix or null matrix or void matrix:

All the entries are zero in the given matrix then the matrix is said ti be a zero  matrix .

Example :

A = `[[0,0,0],[0,0,0],[0,0,0]]` is the null matrix  or zero matrix of order 3 x 3

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Operation on Matrices in math:


Basic operation involved in matrix:

Addition of two matrices in math
Subtraction of matrices in Math
Multiplication of matrices in math.
Let  us discuss about the  operation of matrices in math.

1)Addition of matrices.

Let us consider the two different matrices of order 2 x2

A=`[[3,5],[4,7]]` and B= `[[1,8],[4,7]]`

A+B = `[[3+1,5+8],[4+4,7+7]]`

A+B =`[[4,13],[8,14]]`

Subtraction of two matrices:

Let us consider the two different matrices of order 2 x2

X= `[[4,6],[3,5]]` and Y =`[[5,7],[3,4]]`

X-Y =`[[4-5,6-7],[3-3,5-4]]`

X-Y =`[[-1,-1],[0,1]]`

Multiplication of two matrices:

Let us consider the two different matrices of order 2 x2

P = `[[4,5],[3,2]]` and Q =`[[2,3],[2,1]]`

PQ =`[[4*2 +5*2,4*3+2*1],[3*2+2*2,3*3+2*1]]`

PQ =`[[8+10,12+2],[6+4,9+2]]`

PQ =`[[18,14],[10,11]]`