Chapter 3 MATRICES EXERCISE 3.1 Question
1. In the matrix
2 5 19 -7
35 -2 5/2 12
√3 1 -5 17
write:
(i) The order of the matrix,
(ii) The number of elements,
(iii) Write the elements a13, a21, a33, a24, a23.
Properties of the given matrix:
Question: 1
In the matrix:
2 5 19 -7
35 -2 5/2 12
√3 1 -5 17
(i) The order of the matrix is 3x4.
(ii) The number of elements is 3 * 4 = 12.
(iii) Elements:
- a13 = 19
- a21 = 35
- a33 = -5
- a24 = 12
- a23 = 5/2
2. If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
Possible Matrix Orders
To find the possible orders of a matrix given the number of elements:
- If a matrix has 24 elements:
- 1x24
- 2x12
- 3x8
- 4x6
- If a matrix has 13 elements:
We need to find pairs of factors of 24 to determine possible matrix orders:
So, the possible orders for a matrix with 24 elements are: 1x24, 2x12, 3x8, and 4x6.
There are no pairs of integers whose product is 13, so a matrix with 13 elements cannot exist.
For 13 elements, the only order possible is 1 × 13 or 13 × 1.
3. Possible Matrix Orders
To find the possible orders of a matrix given the number of elements:
- If a matrix has 18 elements:
- 1x18
- 2x9
- 3x6
- If a matrix has 5 elements:
We need to find pairs of factors of 18 to determine possible matrix orders:
So, the possible orders for a matrix with 18 elements are: 1x18, 2x9, and 3x6.
There are no pairs of integers whose product is 5, so a matrix with 5 elements cannot exist.
Question 4. Answer =
(i) Constructing a 2x2 Matrix
To construct a 2x2 matrix A whose elements are given by aij = (i + j)2 / 2:
We can use the following formula for each element:
a11 = (1 + 1)2 / 2 = 2
a12 = (1 + 2)2 / 2 = 2.25
a21 = (2 + 1)2 / 2 = 2.25
a22 = (2 + 2)2 / 2 = 4
So, the matrix A is:
2 2.25
2.25 4
(ii) Constructing a 2x2 Matrix
To construct a 2x2 matrix A whose elements are given by aij = i / j:
We can use the following formula for each element:
a11 = 1 / 1 = 1
a12 = 1 / 2 = 0.5
a21 = 2 / 1 = 2
a22 = 2 / 2 = 1
So, the matrix A is:
1 0.5
2 1
(iii) Constructing a 2x2 Matrix
To construct a 2x2 matrix A whose elements are given by aij = (i + 2j)2 / 2:
We can use the following formula for each element:
a11 = (1 + 2*1)2 / 2 = 9 / 2 = 4.5
a12 = (1 + 2*2)2 / 2 = 25 / 2 = 12.5
a21 = (2 + 2*1)2 / 2 = 16 / 2 = 8
a22 = (2 + 2*2)2 / 2 = 36 / 2 = 18
So, the matrix A is:
4.5 12.5
8 18
Question 5. Answer =
(i) Constructing a 3x4 Matrix
To construct a 3x4 matrix A whose elements are given by aij = 1/2 | -3i + j |:
We can use the following formula for each element:
a11 = 1/2 | -3*1 + 1 | = 1/2 | -2 | = 1
a12 = 1/2 | -3*1 + 2 | = 1/2 | -1 | = 1/2
a13 = 1/2 | -3*1 + 3 | = 1/2 | 0 | = 0
a14 = 1/2 | -3*1 + 4 | = 1/2 | 1 | = 1/2
a21 = 1/2 | -3*2 + 1 | = 1/2 | -5 | = 5/2
a22 = 1/2 | -3*2 + 2 | = 1/2 | -4 | = 2
a23 = 1/2 | -3*2 + 3 | = 1/2 | -3 | = 3/2
a24 = 1/2 | -3*2 + 4 | = 1/2 | -2 | = 1
a31 = 1/2 | -3*3 + 1 | = 1/2 | -8 | = 4
a32 = 1/2 | -3*3 + 2 | = 1/2 | -7 | = 7/2
a33 = 1/2 | -3*3 + 3 | = 1/2 | -6 | = 3
a34 = 1/2 | -3*3 + 4 | = 1/2 | -5 | = 5/2
So, the matrix A is:
1 1/2 0 1/2
5/2 2 3/2 1
4 7/2 3 5/2
(ii) Constructing a 3x4 Matrix
To construct a 3x4 matrix A whose elements are given by aij = 2i - j:
We can use the following formula for each element:
a11 = 2*1 - 1 = 1
a12 = 2*1 - 2 = 0
a13 = 2*1 - 3 = -1
a14 = 2*1 - 4 = -2
a21 = 2*2 - 1 = 3
a22 = 2*2 - 2 = 2
a23 = 2*2 - 3 = 1
a24 = 2*2 - 4 = 0
a31 = 2*3 - 1 = 5
a32 = 2*3 - 2 = 4
a33 = 2*3 - 3 = 3
a34 = 2*3 - 4 = 2
So, the matrix A is:
1 0 -1 -2
3 2 1 0
5 4 3 2
Finding the Values of x, y, and z
To find the values of x, y, and z from the matrix equation:
[ 4 3 ] = [ y z ]
[ x 5 ] [ 1 5 ]
We can equate corresponding elements:
4 = y => y = 4
3 = z => z = 3
x = 1 (from the upper right element of the second matrix)
So, the values of x, y, and z are:
x = 1
y = 4
z = 3
Finding the Values of x, y, and z
To find the values of x, y, and z from the matrix equation:
[ x+y 2 ] = [ 6 2 ]
[ 5+z xy] [ 5 8 ]
We can equate corresponding elements:
find three Equation
x + y = 6 ... (i)
5 + z = 5 , z = 5 - 5 = 0 then z = 0 ...(ii)
xy = 8 ...(iii)
Now We know that
( x - y ) 2 = ( x + y ) 2 - 4xy
( x - y ) 2 = 36 - 32 = 4
x - y = ± 2 ....(iv)
Now x - y = 2 and x + y = 6 then we get x = 4 and y = 2,
When x - y = -2 and x + y = 6 then we get x = 2 and y = 4,
so x = 4, y = 2 and z = 0, or x = 2, y = 4 and z = 0,
(iii)Finding the Values of x, y, and z
To find the values of x, y, and z from the matrix equation:
[ x+y+z ] [ 9 ]
[ x+z ] = [ 5 ]
[ y+z ] [ 7 ]
We can equate corresponding elements:
x + y + z = 9 (1)
x + z = 5 (2)
y + z = 7 (3)
Subtract equation (2) from equation (1) to eliminate x:
(x + y + z) - (x + z) = 9 - 5
y = 4
Substitute y = 4 into equation (3) to find z:
4 + z = 7
z = 3
Substitute z = 3 into equation (2) to find x:
x + 3 = 5
x = 2
So, the values of x, y, and z are:
x = 2
y = 4
z = 3
7. Finding the Values of a, b, c, and d
To find the values of a, b, c, and d from the matrix equation:
[ a-b 2a + c ] = [ -1 5 ]
[ 2a - b 3c + d ] [ 0 13 ]
We can equate corresponding elements:
a - b = -1 (1)
2a + c = 5 (2)
2a - b = 0 (3)
3c + d = 13 (4)
From equations (1) and (3), we can solve for a and b:
From equation (1): a = b - 1
Substitute a into equation (3):
2(b - 1) - b = 0
2b - 2 - b = 0
b - 2 = 0
b = 2
Substitute b = 2 into equation (1):
a - 2 = -1
a = 1
Substitute a = 1 into equation (2) to find c:
2*1 + c = 5
2 + c = 5
c = 3
Substitute c = 3 into equation (4) to find d:
3*3 + d = 13
9 + d = 13
d = 4
So, the values of a, b, c, and d are:
a = 1
b = 2
c = 3
d = 4
8. Square Matrix Definition
A matrix A = [aij] is a square matrix if:
(C) m = n
where m is the number of rows and n is the number of columns.
9. Finding the Values of x and y
To make the pair of matrices equal:
[3x + 7 5 ] = [ 0 y - 2 ]
[ y + 1 2 - 3x ] [ 8 4 ]
We can equate corresponding elements:
3x + 7 = 0 => 3x = -7 => x = -7/3
5 = y - 2 => y = 5 + 2 => y = 7
y + 1 = 8 => y = 8 - 1 => y = 7
2 - 3x = 4 => -3x = 4 - 2 => -3x = 2 => x = -2/3
So, the values of x and y that make the pair of matrices equal are:
x = -7/3 and y = 7
(B) Not possible to find
10. Finding the Number of Possible Matrices
To find the number of possible matrices of order 3 × 3 with each entry 0 or 1:
Each entry in the matrix can be either 0 or 1, so there are 2 choices for each entry.
Therefore, the total number of possible matrices is:
2(3*3) = 29 = 512
So, the correct answer is (D) 512.