NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.1

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BoardCBSE
TextbookNCERT
ClassClass 12
SubjectMaths
ChapterChapter 3
Chapter NameMatrices
ExerciseEx 3.1
Number of Questions Solved10
CategoryNCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.1

Ex 3.1 Class 12 Maths Question 1.
In the matrix
(i) The order of the matrix
(ii) The number of elements
(iii) Write the elements a13, a21, a33, a24, a23
Solution:
(i) The matrix A has three rows and 4 columns.
The order of the matrix is 3 x 4.
(ii) There are 3 x 4 = 12 elements in the matrix A
(iii) a13 = 19, a21 = 35, a33 = – 5, a24 = 12, a23 = \ frac { 5 }{ 2 }

Ex 3.1 Class 12 Maths Question 2.
If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
Solution:
(i) 24 = 1 x 24 = 2 x 12 = 3 x 8 = 4 x 6
Thus there are 8 matrices having 24 elements their order are (1 x 24), (24 x 1), (2 x 12), (12 x 2),(3 x 8), (8 x 3), (4 x 6), (6 x 4).
(ii) 13 = 1 x 13,
There are 2 matrices of 13 elements of order (1 x 13) and (13 x 1).

Ex 3.1 Class 12 Maths Question 3.
If a matrix has 18 elements, what are the possible orders it can have ? What, if it has 5 elements.
Solution:
We know that if a matrix is of order m × n, it has mn elements.
=> 18 = 1 x 18 = 2 x 9 = 3 x 6
Thus, all possible ordered pairs of the matrix
having 18 elements are:
(1,18), (18,1), (2,9), (9,2), (3,6), (6,3)
If it has 5 elements, then possible order are: (1,5), (5,1)

Ex 3.1 Class 12 Maths Question 4.
Construct a 2 x 2 matrix, A= [aij] whose elements are given by:



Solution:
A={ left[ { a }_{ ij } right] }_{ 2times 2 }=begin{bmatrix} { a }_{ 11 } & { a }_{ 12 } \ { a }_{ 21 } & { a }_{ 22 } end{bmatrix}
(i)quad { a }_{ ij }=frac { { (i+j) }^{ 2 } }{ 2 }
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 4
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 4.1

Ex 3.1 Class 12 Maths Question 5.
Construct a 3 x 4 matrix , whose elements are given by:


Solution:
A={ left[ { a }_{ ij } right] }_{ 3times 4 }=left[ begin{matrix} { a }_{ 11 } \ { a }_{ 21 } \ { a }_{ 31 } end{matrix}begin{matrix} { a }_{ 12 } \ { a }_{ 22 } \ { a }_{ 32 } end{matrix}begin{matrix} { a }_{ 13 } \ { a }_{ 23 } \ { a }_{ 33 } end{matrix}begin{matrix} { a }_{ 14 } \ { a }_{ 24 } \ { a }_{ 34 } end{matrix} right]
(i){ a }_{ ij }=frac { 1 }{ 2 } left| -3i+j right|
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 5
byjus class 12 maths Chapter 3 Matrices 5.1

Ex 3.1 Class 12 Maths Question 6.
Find the values of x, y, z from the following equations:
(i)begin{bmatrix} 4 & 3 \ x & 5 end{bmatrix}=begin{bmatrix} y & z \ 1 & 5 end{bmatrix}
(ii)begin{bmatrix} x+y & 2 \ 5+z & xy end{bmatrix}=begin{bmatrix} 6 & 2 \ 5 & 8 end{bmatrix}
(iii)left[ begin{matrix} begin{matrix} x+ & y+ & z end{matrix} \ begin{matrix} x & +y end{matrix} \ begin{matrix} y & +z end{matrix} end{matrix} right] =left[ begin{matrix} 9 \ 5 \ 7 end{matrix} right]
Solution:
(i)begin{bmatrix} 4 & 3 \ x & 5 end{bmatrix}=begin{bmatrix} y & z \ 1 & 5 end{bmatrix}
Clearly x = 1,y = 4,z = 3
(ii)begin{bmatrix} x+y & 2 \ 5+z & xy end{bmatrix}=begin{bmatrix} 6 & 2 \ 5 & 8 end{bmatrix}
Now 5 + z = 5 => z = 0
Now x + y = 6 and xy = 8
∴ y = 6 – x and x(6 – x) = 8
6x – x² = 8
x² – 6x + 8 = 0
(x – 4)(x – 2) = 0
=>x = 2,4
When x = 2, y = 6 – 2 = 4
and when x = 4,y = 6 – 4 = 2
Hence x = 2,y = 4,z = 0 or x = 4,y = 2,z = 0.
(iii) Equating the corresponding elements.
=> x+y+z=9 …..(i)
x+z = 5 …(ii)
y+ z = 7 …(iii)
Adding eqs. (ii) & (iii)
x + y + 2z = 12
=> (x+y+z) + z = 12,
9+z = 12 (from equ (i))
z = 3
x + z = 5
=>x + 3 = 5 => x = 2
and y+z = 7
=>y+3 = 7
=> y = 4
=> x = 2, y = 4 and z = 3

Ex 3.1 Class 12 Maths Question 7.
Find the values of a,b,c and d from the equation:
begin{bmatrix} a-b & 2a+c \ 2a-b & 3c+d end{bmatrix}=begin{bmatrix} -1 & 5 \ 0 & 13 end{bmatrix}
Solution:
begin{bmatrix} a-b & 2a+c \ 2a-b & 3c+d end{bmatrix}=begin{bmatrix} -1 & 5 \ 0 & 13 end{bmatrix}
byjus class 12 maths Chapter 3 Matrices 7

Ex 3.1 Class 12 Maths Question 8.
A = [aij]m×n is a square matrix, if
(a) m < n (b) n > n
(c) m = n
(d) none of these
Solution:
For a square matrix m=n.
Thus option (c) m = n, is correct.

Ex 3.1 Class 12 Maths Question 9.
Which of the given values of x and y make the following pairs of matrices equal:

(a)
(b) Not possible to find
(c)
(d)
Solution:
begin{bmatrix} 3x+7 & 5 \ y+1 & 2-3x end{bmatrix},begin{bmatrix} 0 & y-2 \ 8 & 4 end{bmatrix}
(a) x=frac { -1 }{ 3 } ,y=7
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 9
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 9.1

Ex 3.1 Class 12 Maths Question 10.
The number of all possible matrices of order 3×3 with each entry 0 or 1 is
(a) 27
(b) 18
(c) 81
(d) 512
Solution:
There are 3 x 3 matrix or 9 entries in matrix each place can be filled with 0 or 1
∴ 9 Places can be filled in 29 = 512 ways
Number of such matrices = 512
Option (d) is correct.

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