Matrices(Ex. 3.1)

Exercise 3.1

1.  

    

(i) The order of the matrix,

Solution:- No. of rows = 3

No. of columns = 4

So, the order of the matrix is = 3x4. (Ans....)

(ii) The number of elements,

Solution:- No. of elements = 12 (Ans....)

(iii) Write the element a13, a21, a33, a24, a23.

Solution:- a13=19 

a21=35 ; a33=-5 ; a24=12 ; a23=5/2 (Ans....)

2. If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

Solun:- If matrix has 24 elements then its order is:-

= 1x24

= 2x12

= 3x8

= 4x6

= 6x4

= 8x3

= 12x2

= 24x1

If matrix has 13 elements then order is:-

= 1x13

= 13x1

3. If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Solun:- If matrix has 18 elements then its order is:-

= 1x18

= 2x9

= 3x6

= 6x3

= 9x2

= 18x1

If matrix has 5 elements then its order is:-

= 1x5

= 5x1

4. Construct 2x2 matrix, A = [aij], whose elements are given by:

(i) aij = (i+j)2/2

Solun:- Let 2x2 order matrix is:-

       

Elements are:-

a11 = (1+1)2/2 = 4/2 = 2      

a12  = (1+2)2/2 = 9/2 

a21 = (2+1)2/2 = 9/2

a22 = (2+2)2/2 = 16/2 = 8  

         (Ans....)

(ii) aij = i / j

Solun:- Let 2x2 order matrix is:-

       

Elements are:- 

a11 = 1/1 = 1

a12 = 1/2    

a21 = 2/1 = 2

a22 = 2/2 = 1

        (Ans....)

(iii) aij = (i+2j)2/2 

Solun:- Let 2x2 order matrix is:-

       

Elements are:- 

a11 = (1+2x1)2/2 = 9/2

a12 = (1+2x2)2/2 = 25/2

a21 = (2+2x1)2/2 = 8

a22 = (2+2x2)2/2 = 18

       (Ans....)

5. Construct 3x4 matrix, whose elements are given by:

(i) aij = |-3i+j| / 2

Solun:- Let 3x4 order matrix is:-

     

Elements are:-

a11 = |-3x1+1|/2 = 1

a12 = |-3x1+2|/2 = 1/2

a13 = |-3x1+3|/2 = 0

a14 = |-3x1+4|/2 = 1/2

a21 = |-3x2+1|/2 = 5/2

a22 = |-3x2+2|/2 = 2

a23 = |-3x2+3|/2 = 3/2

a24 = |-3x2+4|/2 = 1

a31 = |-3x3+1|/2 = 4

a32 = |-3x3+2|/2 = 7/2

a33 = |-3x3+3|/2 = 3

a34 = |-3x3+4|/2 = 5/2

     (Ans....)

(ii) aij = 2i - j

Solun:- Let 3x4 order matrix is:-

     

Elements are:-

a11 = 2x1-1 = 1

a12 = 2x1-2 = 0

a13 = 2x1-3 = -1

a14 = 2x1-4 = -2

a21 = 2x2-1 = 3

a22 = 2x2-2 = 2

a23 = 2x2-3 = 1

a24 = 2x2-4 = 0

a31 = 2x3-1 = 5

a32 = 2x3-2 = 4

a33 = 2x3-3 = 3

a34 = 2x3-4 = 2

    (Ans....)

6. Find the values of x, y, and z from the following equations:

 

Solun:- Comparing the corresponding elements of matrices:-

⇒ y = 4 ; z = 3 ; x = 1  

Solun:- Comparing the corresponding elements of matrices:-

⇒ x+y = 6……(1)

⇒ xy = 8

⇒ y = 8/x

Put the value of y in Eq. 1:-

⇒ x+(8/x) = 6

⇒ x2+8 = 6x

⇒ x2-6x+8 = 0

⇒ x2-4x-2x+8 = 0

⇒ x(x-4)-2(x-4) =0

⇒ (x-4)(x-2) = 0

⇒ x = 4,2 then y = 2,4

⇒ 5+z = 5

⇒ z = 0

⇒ x = 4, y=2, z=0 or x=2, y=4, z=0  (Ans....)

Solun:- Comparing the corresponding elements of matrices:- 

⇒ x+y+z = 9……(1)

⇒ x+z = 5……(2)

⇒ y+z = 7……(3)

Put the value of Eq. 3 in Eq. 1:-

⇒ x+7 = 9

⇒ x = 2

Put the value of x in Eq. 2:-

⇒ 2+z = 5

⇒ z = 3

Put the value of z in Eq. 3:-

⇒ y+3 = 7

⇒ y = 4

⇒ x = 2, y = 4, z = 3 (Ans....)

7. Find the value of a, b, c and d from the equation:

    

Solun:- Comparing the corresponding elements of matrices:-

⇒ a-b = -1……(1)

⇒ 2a+c = 5……(2)

⇒ 2a-b = 0……(3)

⇒ 3c+d = 13……(4)

Solving these equations:-

From Eq. 1 and 3:- apply elimination method

⇒ a = 1, b = 2

Put the value of a in Eq. 2:- 

⇒ 2+c = 5

⇒ c = 3

Put the value of c in Eq. 4:-

⇒ 9+d = 13

⇒ d = 4

⇒ a=1, b= 2, c=3 and d= 4 (Ans....)

8. A = [aij]mxn is a square matrix, if

(A) m<n 

(B) m>n

(C) m=n

(D) None of these

Solun:- By definition of square matrix:-

Number of rows = Number of columns of the matrix

⇒ m=n

Answer is C.

9. Which of the given values of x and y make the following pair of matrices equal

    

(A) x = -1/3 , y=7

(B) Not possible to find
(C) y = 7 , x = -2/3
(D) x = -1/3 , y = -⅔

Solution:- Given that matrices are equal then

 

⇒ 3x+7 = 0; x = -7/3

⇒ y-2 = 5; y = 7

⇒ y+1 = 8; y = 7

⇒ 2-3x = 4; x = -2/3

Variable x has two values but that is not possible.

So, matrices are not equal.

Answer is B.

10. The number of all possible matrices of order 3x3 with each entry 0 or 1 is:

(A) 27
(B) 18
(C) 81
(D) 512

Solution:- The number of all possible matrices of order 3x3 with each entry 0 or 1 is:

Then there are 9 elements in the matrix and each element has two choices = 29 = 512.

Answer is D.

Download Exercise 3.1 in PDF

Download Definitions and formulas of Matrices

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