RD Sharma Class 12 Solutions - Algebra of Matrices | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


5. Algebra of Matrices
Exercise 5.1
1. Question
If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?
Answer
If a matrix is of order m×n elements, it has mn elements. So, if the matrix has 8 elements, we will find the
ordered pairs m and n.
mn = 8
Then, ordered pairs m and n can be
m×n be (8×1),(1×8),(4×2),(2×4)
Now, if it has 5 elements
Possible orders are (5×1), (1×5).
2 A. Question
If and then find
a
22
 + b
21
Answer
A = [a
ij
] =  …......(1)
B = [b
ij
] =  …......(2)
Given, A = [a
ij
] =  B = [b
ij
] = 
Now, Comparing with equation (1) and (2)
a
22
 = 4 and b
21
 = – 3
a
22
 + b
21
 = 4 + ( – 3) = 1
2 B. Question
If and then find
a
11
 b
11
 + a
22
 b
22
Answer
A = [a
ij
] =  …......(1)
Page 2


5. Algebra of Matrices
Exercise 5.1
1. Question
If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?
Answer
If a matrix is of order m×n elements, it has mn elements. So, if the matrix has 8 elements, we will find the
ordered pairs m and n.
mn = 8
Then, ordered pairs m and n can be
m×n be (8×1),(1×8),(4×2),(2×4)
Now, if it has 5 elements
Possible orders are (5×1), (1×5).
2 A. Question
If and then find
a
22
 + b
21
Answer
A = [a
ij
] =  …......(1)
B = [b
ij
] =  …......(2)
Given, A = [a
ij
] =  B = [b
ij
] = 
Now, Comparing with equation (1) and (2)
a
22
 = 4 and b
21
 = – 3
a
22
 + b
21
 = 4 + ( – 3) = 1
2 B. Question
If and then find
a
11
 b
11
 + a
22
 b
22
Answer
A = [a
ij
] =  …......(1)
B = [b
ij
] =  …......(2)
Given, A = [a
ij
] =  B = [b
ij
] = 
Now, Comparing with equation (1) and (2)
a
11
 = 2, a
22
 = 4, b
11
 = 2, b
22
 = 4
a
11
 b
11
 + a
22
 b
22
 = 2 × 2 + 4 × 4 = 4 + 16 = 20
3. Question
Let A be a matrix of order 3 × 4. If R
1
 denotes the first row of A and C
2
 denotes its second column, then
determine the orders of matrices R
1
 and C
2
.
Answer
Let A be a matrix of order 3×4.
A = [a
ij
]
3×4
R
1
 = first row of A = [a
11
,a
12
,a
13
,a
14
]
So, order of matrix R
1
 = 1×4
C
2
 = second column of A = 
Order of C
2
 = 3×1
4 A. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = i × j
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A = ………………. (1)
a
11
 = 1×1 = 1 a
12
 = 1×2 = 2 a
13
 = 1×3 = 3
a
21
 = 2×1 = 2 a
22
 = 2×2 = 4 a
23
 = 2×3 = 6
So, from (1)
A = 
4 B. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = 2i – j
Answer
Page 3


5. Algebra of Matrices
Exercise 5.1
1. Question
If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?
Answer
If a matrix is of order m×n elements, it has mn elements. So, if the matrix has 8 elements, we will find the
ordered pairs m and n.
mn = 8
Then, ordered pairs m and n can be
m×n be (8×1),(1×8),(4×2),(2×4)
Now, if it has 5 elements
Possible orders are (5×1), (1×5).
2 A. Question
If and then find
a
22
 + b
21
Answer
A = [a
ij
] =  …......(1)
B = [b
ij
] =  …......(2)
Given, A = [a
ij
] =  B = [b
ij
] = 
Now, Comparing with equation (1) and (2)
a
22
 = 4 and b
21
 = – 3
a
22
 + b
21
 = 4 + ( – 3) = 1
2 B. Question
If and then find
a
11
 b
11
 + a
22
 b
22
Answer
A = [a
ij
] =  …......(1)
B = [b
ij
] =  …......(2)
Given, A = [a
ij
] =  B = [b
ij
] = 
Now, Comparing with equation (1) and (2)
a
11
 = 2, a
22
 = 4, b
11
 = 2, b
22
 = 4
a
11
 b
11
 + a
22
 b
22
 = 2 × 2 + 4 × 4 = 4 + 16 = 20
3. Question
Let A be a matrix of order 3 × 4. If R
1
 denotes the first row of A and C
2
 denotes its second column, then
determine the orders of matrices R
1
 and C
2
.
Answer
Let A be a matrix of order 3×4.
A = [a
ij
]
3×4
R
1
 = first row of A = [a
11
,a
12
,a
13
,a
14
]
So, order of matrix R
1
 = 1×4
C
2
 = second column of A = 
Order of C
2
 = 3×1
4 A. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = i × j
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A = ………………. (1)
a
11
 = 1×1 = 1 a
12
 = 1×2 = 2 a
13
 = 1×3 = 3
a
21
 = 2×1 = 2 a
22
 = 2×2 = 4 a
23
 = 2×3 = 6
So, from (1)
A = 
4 B. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = 2i – j
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A = ………………. (1)
a
11
 = 2×1 – 1 = 2 – 1 = 1 a
12
 = 2×1 – 2 = 2 – 2 = 0 a
13
 = 2×1 – 3 = 2 – 3 = – 1
a
21
 = 2×2 – 1 = 4 – 1 = 3 a
22
 = 2×2 – 2 = 4 – 2 = 2 a
23
 = 2×2 – 3 = 4 – 3 = 1
So, from (1)
A = 
4 C. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = i + j
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A =  …… (1)
a
11
 = 1 + 1 = 2 a
12
 = 1 + 2 = 3 a
13
 = 1 + 3 = 4
a
21
 = 2 + 1 = 3 a
22
 = 2 + 2 = 4 a
23
 = 2 + 3 = 5
So, from (1)
A = 
4 D. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = 
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A =  …… (1)
a
11
 = 
a
12
 = 
a
13
 = 
Page 4


5. Algebra of Matrices
Exercise 5.1
1. Question
If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?
Answer
If a matrix is of order m×n elements, it has mn elements. So, if the matrix has 8 elements, we will find the
ordered pairs m and n.
mn = 8
Then, ordered pairs m and n can be
m×n be (8×1),(1×8),(4×2),(2×4)
Now, if it has 5 elements
Possible orders are (5×1), (1×5).
2 A. Question
If and then find
a
22
 + b
21
Answer
A = [a
ij
] =  …......(1)
B = [b
ij
] =  …......(2)
Given, A = [a
ij
] =  B = [b
ij
] = 
Now, Comparing with equation (1) and (2)
a
22
 = 4 and b
21
 = – 3
a
22
 + b
21
 = 4 + ( – 3) = 1
2 B. Question
If and then find
a
11
 b
11
 + a
22
 b
22
Answer
A = [a
ij
] =  …......(1)
B = [b
ij
] =  …......(2)
Given, A = [a
ij
] =  B = [b
ij
] = 
Now, Comparing with equation (1) and (2)
a
11
 = 2, a
22
 = 4, b
11
 = 2, b
22
 = 4
a
11
 b
11
 + a
22
 b
22
 = 2 × 2 + 4 × 4 = 4 + 16 = 20
3. Question
Let A be a matrix of order 3 × 4. If R
1
 denotes the first row of A and C
2
 denotes its second column, then
determine the orders of matrices R
1
 and C
2
.
Answer
Let A be a matrix of order 3×4.
A = [a
ij
]
3×4
R
1
 = first row of A = [a
11
,a
12
,a
13
,a
14
]
So, order of matrix R
1
 = 1×4
C
2
 = second column of A = 
Order of C
2
 = 3×1
4 A. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = i × j
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A = ………………. (1)
a
11
 = 1×1 = 1 a
12
 = 1×2 = 2 a
13
 = 1×3 = 3
a
21
 = 2×1 = 2 a
22
 = 2×2 = 4 a
23
 = 2×3 = 6
So, from (1)
A = 
4 B. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = 2i – j
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A = ………………. (1)
a
11
 = 2×1 – 1 = 2 – 1 = 1 a
12
 = 2×1 – 2 = 2 – 2 = 0 a
13
 = 2×1 – 3 = 2 – 3 = – 1
a
21
 = 2×2 – 1 = 4 – 1 = 3 a
22
 = 2×2 – 2 = 4 – 2 = 2 a
23
 = 2×2 – 3 = 4 – 3 = 1
So, from (1)
A = 
4 C. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = i + j
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A =  …… (1)
a
11
 = 1 + 1 = 2 a
12
 = 1 + 2 = 3 a
13
 = 1 + 3 = 4
a
21
 = 2 + 1 = 3 a
22
 = 2 + 2 = 4 a
23
 = 2 + 3 = 5
So, from (1)
A = 
4 D. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = 
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A =  …… (1)
a
11
 = 
a
12
 = 
a
13
 = 
a
21
 = 
a
22
 = 
a
23
 = 
So, from (1)
A = 
5 A. Question
Construct a 2 × 2 matrix A = [a
jj
] whose elements a
jj
 are given by :
Answer
Let A = [a
ij
]
2×2
So, the elements in a 2×2 matrix are
a
11
, a
12
, a
21
, a
22
,
A =  …… (1)
a
11
 = 
a
12
 = 
a
21
 = 
a
22
 = 
So, from (1)
A = 
5 B. Question
Construct a 2 × 2 matrix A = [a
jj
] whose elements a
jj
 are given by :
Answer
Let A = [a
ij
]
2×2
So, the elements in a 2×2 matrix are
a
11
, a
12
, a
21
, a
22
,
A =  …… (1)
a
11
 = 
Page 5


5. Algebra of Matrices
Exercise 5.1
1. Question
If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?
Answer
If a matrix is of order m×n elements, it has mn elements. So, if the matrix has 8 elements, we will find the
ordered pairs m and n.
mn = 8
Then, ordered pairs m and n can be
m×n be (8×1),(1×8),(4×2),(2×4)
Now, if it has 5 elements
Possible orders are (5×1), (1×5).
2 A. Question
If and then find
a
22
 + b
21
Answer
A = [a
ij
] =  …......(1)
B = [b
ij
] =  …......(2)
Given, A = [a
ij
] =  B = [b
ij
] = 
Now, Comparing with equation (1) and (2)
a
22
 = 4 and b
21
 = – 3
a
22
 + b
21
 = 4 + ( – 3) = 1
2 B. Question
If and then find
a
11
 b
11
 + a
22
 b
22
Answer
A = [a
ij
] =  …......(1)
B = [b
ij
] =  …......(2)
Given, A = [a
ij
] =  B = [b
ij
] = 
Now, Comparing with equation (1) and (2)
a
11
 = 2, a
22
 = 4, b
11
 = 2, b
22
 = 4
a
11
 b
11
 + a
22
 b
22
 = 2 × 2 + 4 × 4 = 4 + 16 = 20
3. Question
Let A be a matrix of order 3 × 4. If R
1
 denotes the first row of A and C
2
 denotes its second column, then
determine the orders of matrices R
1
 and C
2
.
Answer
Let A be a matrix of order 3×4.
A = [a
ij
]
3×4
R
1
 = first row of A = [a
11
,a
12
,a
13
,a
14
]
So, order of matrix R
1
 = 1×4
C
2
 = second column of A = 
Order of C
2
 = 3×1
4 A. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = i × j
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A = ………………. (1)
a
11
 = 1×1 = 1 a
12
 = 1×2 = 2 a
13
 = 1×3 = 3
a
21
 = 2×1 = 2 a
22
 = 2×2 = 4 a
23
 = 2×3 = 6
So, from (1)
A = 
4 B. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = 2i – j
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A = ………………. (1)
a
11
 = 2×1 – 1 = 2 – 1 = 1 a
12
 = 2×1 – 2 = 2 – 2 = 0 a
13
 = 2×1 – 3 = 2 – 3 = – 1
a
21
 = 2×2 – 1 = 4 – 1 = 3 a
22
 = 2×2 – 2 = 4 – 2 = 2 a
23
 = 2×2 – 3 = 4 – 3 = 1
So, from (1)
A = 
4 C. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = i + j
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A =  …… (1)
a
11
 = 1 + 1 = 2 a
12
 = 1 + 2 = 3 a
13
 = 1 + 3 = 4
a
21
 = 2 + 1 = 3 a
22
 = 2 + 2 = 4 a
23
 = 2 + 3 = 5
So, from (1)
A = 
4 D. Question
Construct a 2 ×3 matrix A = [a
jj
] whose elements a
jj
 are given by :
a
ij
 = 
Answer
Let A = [a
ij
]
2×3
So, the elements in a 2×3 matrix are
a
11
, a
12
, a
13
, a
21
, a
22
, a
23
A =  …… (1)
a
11
 = 
a
12
 = 
a
13
 = 
a
21
 = 
a
22
 = 
a
23
 = 
So, from (1)
A = 
5 A. Question
Construct a 2 × 2 matrix A = [a
jj
] whose elements a
jj
 are given by :
Answer
Let A = [a
ij
]
2×2
So, the elements in a 2×2 matrix are
a
11
, a
12
, a
21
, a
22
,
A =  …… (1)
a
11
 = 
a
12
 = 
a
21
 = 
a
22
 = 
So, from (1)
A = 
5 B. Question
Construct a 2 × 2 matrix A = [a
jj
] whose elements a
jj
 are given by :
Answer
Let A = [a
ij
]
2×2
So, the elements in a 2×2 matrix are
a
11
, a
12
, a
21
, a
22
,
A =  …… (1)
a
11
 = 
a
12
 = 
a
21
 = 
a
22
 = 
So, from (1)
A = 
5 C. Question
Construct a 2 × 2 matrix A = [a
jj
] whose elements a
jj
 are given by :
Answer
Let A = [a
ij
]
2×2
So, the elements in a 2×2 matrix are
a
11
, a
12
, a
21
, a
22
,
A =  …… (1)
a
11
 = 
a
12
 = 
a
21
 = 
a
22
 = 
So, from (1)
A = 
5 D. Question
Construct a 2 × 2 matrix A = [a
jj
] whose elements a
jj
 are given by :
Answer
Let A = [a
ij
]
2×2
So, the elements in a 2×2 matrix are
a
11
, a
12
, a
21
, a
22
,
A =  …… (1)
a
11
 =