Class 8 – Matrices

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Introduction: The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. The evolution of concept of matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations.

 

Matrix: A matrix is an ordered rectangular array of numbers arranged in rows and columns

Example:

Knowledge of matrices:

I. Plural of matrix is matrices. Each number or entity in matrix is called its element. In a matrix, the horizontal lines are called rows; whereas the vertical lines are called

columns.

Example: In  2, 3, 7 from a row where as 2, 5 from a column.

 

II. Order of a matrix: The order of a matrix
= Number of rows in it × Number of columns in it;
i.e. if a matrix has m number of rows and n number of columns, its order is written as  and is read as m by n.

It has 2 rows and 3 columns; hence its order (read as 2 by 3)

While stating the order of a matrix, the number of rows is given first and then the number of columns.

 

III. Representation of a matrix:

Matrices, in general, are denoted by capital letters. For example, if A is a matrix with

m rows and n columns, then it is written as a  similarly,   means, a matrix B with 2 rows and 3 columns.

 

IV. Elements of a matrix: Each number or entity in a matrix is called its element. The total number of elements in a matrix is equal to the product of its number of rows and number of columns, i.e. if a matrix has 4 rows and 6 columns then the number of elements in it = 4 × 6 = 24

Since, matrix A has 2 rows and 3 columns. So, the number of elements in it

V.  Types of matrices:

a)   Row matrix: A matrix which has only one row is called a row matrix.
Example:

This matrix has 1 row and 2 columns.

b)   Column Matrix: A matrix which has only one column is called a column matrix.

Since, this matrix has 2 rows and 1 column, its order =

c)   Square matrix: A matrix which has an equal number of rows and columns is called a square matrix.
Example:        

Since,the matrix has 2 rows and 2 columns, its order = 2 × 2 (2 by 2).

Similarly, is a square matrix of order 3 × 3

d)   Rectangular Matrix: A matrix in which the number of rows are not equal to the number of columns is called a rectangular matrix.
Example:

e)   Zero or Null matrix: If each element of a matrix is zero is called as zero matrix or null matrix. It is denoted by ‘0’.
Example:

f)    Diagonal Matrix: A square matrix which has all its elements zero, except those on the leading (or) principal diagonal is called a diagonal matrix.
Example:

http://images.clipartpanda.com/bloc-clipart-note-md.pngIn a square matrix the elements of first row first one, second row second one and so on are called principal diagonal elements.

g)   Unit or Identity Matrix: A diagonal matrix in which each element of its leading diagonal is unity is called a unit or identity matrix. It is denoted by I. In other words, it is a square matrix in which each element of its leading diagonal is equal to 1 and all other remaining elements of the matrix are zero.
Example:

VI. Transpose of a matrix: Transpose of a matrix is the matrix obtained by interchanging its rows and columns.

If A is a matrix, then its transpose is denoted by

Example:

VII. Equality of Matrices: Two matrices are said to be equal if:

(i)Both the matrices have the same order,

(ii)The corresponding elements of both the matrices are equal.

Example: Find the values of x, y, a and b, if:

Solution:

Addition of Matrices:

Compatibility for addition of matrices: Two matrices can be added together, if they are of the same order.

To add two matrices of the same order means to add the corresponding elements of both the matrices.

Example:

Solution:

Negative of a Matrix: The negative of a matrix A denoted by  is the matrix formed by replacing each entry in matrix A with the additive inverse of it.

The sum of a matrix and its negative is always a zero matrix.

Subtraction of Matrices: The same rule (or) method is used for the subtraction of matrices, which is used for the addition of matrices.

Example: then find A – B.

Solution:

Example:

What do you observe?

Solution:

We observe that .

(ii)

   

We observe that.

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1.  In addition or subtraction of the matrices, the order of the resulting matrix is the same as the order of matrices added or subtracted.

2.  If A, B and C are the matrices of the same order, then:
(i)                           i.e.addition of matrices is commutative.
(ii)        i.e.addition of matrices is associative.
(iii)           i.e., solving for X matrix