NCERT Solutions for Class 12 Science Math Chapter 3 - Matrices

Answer:

(i) In the given matrix, the number of rows is 3 and the number of columns is 4. Therefore, the order of the matrix is 3 × 4.

(ii) Since the order of the matrix is 3 × 4, there are 3 × 4 = 12 elements in it.

(iii) a₁₃ = 19, a₂₁ = 35, a₃₃ = −5, a₂₄ = 12, a₂₃ =

Answer:

We know that if a matrix is of the order m × n, it has mn elements. Thus, to find all the possible orders of a matrix having 24 elements, we have to find all the ordered pairs of natural numbers whose product is 24.

The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and
(6, 4)

Hence, the possible orders of a matrix having 24 elements are:

1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, and 6 × 4

(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13.

Hence, the possible orders of a matrix having 13 elements are 1 × 13 and 13 × 1.

Answer:

We know that if a matrix is of the order m × n, it has mn elements. Thus, to find all the possible orders of a matrix having 18 elements, we have to find all the ordered pairs of natural numbers whose product is 18.

The ordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6,), and (6, 3)

Hence, the possible orders of a matrix having 18 elements are:

1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, and 6 × 3

(1, 5) and (5, 1) are the ordered pairs of natural numbers whose product is 5.

Hence, the possible orders of a matrix having 5 elements are 1 × 5 and 5 × 1.

Answer:

In general, a 2 × 2 matrix is given by

(i)

Therefore, the required matrix is

(ii)

Therefore, the required matrix is

(iii)

Therefore, the required matrix is

Answer:

In general, a 3 × 4 matrix is given by

(i)

Therefore, the required matrix is

(ii)

Therefore, the required matrix is

Answer:

(i)

As the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

x = 1, y = 4, and z = 3

(ii)

As the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

x + y = 6, xy = 8, 5 + z = 5

Now, 5 + z = 5 ⇒ z = 0

We know that:

(x − y)² = (x + y)² − 4xy

⇒ (x − y)² = 36 − 32 = 4

⇒ x − y = ±2

Now, when x − y = 2 and x + y = 6, we get x = 4 and y = 2

When x − y = − 2 and x + y = 6, we get x = 2 and y = 4

∴x = 4, y = 2, and z = 0 or x = 2, y = 4, and z = 0

(iii)

As the two matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

x + y + z = 9 … (1)

x + z = 5 … (2)

y + z = 7 … (3)

From (1) and (2), we have:

y + 5 = 9

⇒ y = 4

Then, from (3), we have:

4 + z = 7

⇒ z = 3

∴ x + z = 5

⇒ x = 2

∴ x = 2, y = 4, and z = 3

Answer:

As the two matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

a − b = −1 … (1)

2a − b = 0 … (2)

2a + c = 5 … (3)

3c + d = 13 … (4)

From (2), we have:

b = 2a

Then, from (1), we have:

a − 2a = −1

⇒ a = 1

⇒ b = 2

Now, from (3), we have:

2 ×1 + c = 5

⇒ c = 3

From (4) we have:

3 ×3 + d = 13

⇒ 9 + d = 13 ⇒ d = 4

∴a = 1, b = 2, c = 3, and d = 4

Answer:

The correct answer is C.

It is known that a given matrix is said to be a square matrix if the number of rows is equal to the number of columns.

Therefore, is a square matrix, if m = n.

Answer:

The correct answer is B.

It is given that

Equating the corresponding elements, we get:

We find that on comparing the corresponding elements of the two matrices, we get two different values of x, which is not possible.

Hence, it is not possible to find the values of x and y for which the given matrices are equal.

Answer:

The correct answer is D.

The given matrix of the order 3 × 3 has 9 elements and each of these elements can be either 0 or 1.

Now, each of the 9 elements can be filled in two possible ways.

Therefore, by the multiplication principle, the required number of possible matrices is 2⁹ = 512

Answer:

(i)

(ii)

(iii)

(iv) Matrix A has 2 columns. This number is equal to the number of rows in matrix B. Therefore, AB is defined as:

(v) Matrix B has 2 columns. This number is equal to the number of rows in matrix A. Therefore, BA is defined as:

Answer:

(i)

(ii)

(iii)

(iv)

Answer:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Answer:

Answer:

Answer:

Answer:

(i)

Adding equations (1) and (2), we get:

(ii)

Multiplying equation (3) with (2), we get:

Multiplying equation (4) with (3), we get:

From (5) and (6), we have:

Now,

Answer:

Answer:

Comparing the corresponding elements of these two matrices, we have:

∴x = 3 and y = 3

Answer:

Comparing the corresponding elements of these two matrices, we get:

Answer:

Comparing the corresponding elements of these two matrices, we get:

2x − y = 10 and 3x + y = 5

Adding these two equations, we have:

5x = 15

⇒ x = 3

Now, 3x + y = 5

⇒ y = 5 − 3x

⇒ y = 5 − 9 = −4

∴x = 3 and y = −4

Answer:

Comparing the corresponding elements of these two matrices, we get:

Answer:

Answer:

(i)

(ii)

Answer:

We have A² = A × A

Answer:

Answer:

Comparing the corresponding elements, we have:

Thus, the value of k is 1.

Answer:

Answer:

(a) Let Rs x be invested in the first bond. Then, the sum of money invested in the second bond will be Rs (30000 − x).

It is given that the first bond pays 5% interest per year and the second bond pays 7% interest per year.

Therefore, in order to obtain an annual total interest of Rs 1800, we have:

Thus, in order to obtain an annual total interest of Rs 1800, the trust fund should invest Rs 15000 in the first bond and the remaining Rs 15000 in the second bond.

(b) Let Rs x be invested in the first bond. Then, the sum of money invested in the second bond will be Rs (30000 − x).

Therefore, in order to obtain an annual total interest of Rs 2000, we have:

Thus, in order to obtain an annual total interest of Rs 2000, the trust fund should invest Rs 5000 in the first bond and the remaining Rs 25000 in the second bond.

Answer:

The bookshop has 10 dozen chemistry books, 8 dozen physics books, and 10 dozen economics books.

The selling prices of a chemistry book, a physics book, and an economics book are respectively given as Rs 80, Rs 60, and Rs 40.

The total amount of money that will be received from the sale of all these books can be represented in the form of a matrix as:

Thus, the bookshop will receive Rs 20160 from the sale of all these books.

Answer:

Matrices P and Y are of the orders p × k and 3 × k respectively.

Therefore, matrix PY will be defined if k = 3. Consequently, PY will be of the order p × k.

Matrices W and Y are of the orders n × 3 and 3 × k respectively.

Since the number of columns in W is equal to the number of rows in Y, matrix WY is well-defined and is of the order n × k.

Matrices PY and WY can be added only when their orders are the same.

However, PY is of the order p × k and WY is of the order n × k. Therefore, we must have p = n.

Thus, k = 3 and p = n are the restrictions on n, k, and p so that will be defined.

Answer:

The correct answer is B.

Matrix X is of the order 2 × n.

Therefore, matrix 7X is also of the same order.

Matrix Z is of the order 2 × p, i.e., 2 × n [Since n = p]

Therefore, matrix 5Z is also of the same order.

Now, both the matrices 7X and 5Z are of the order 2 × n.

Thus, matrix 7X − 5Z is well-defined and is of the order 2 × n.

Answer:

(i)

(ii)

(iii)

Answer:

We have:

(i)

(ii)

Answer:

(i) It is known that

Therefore, we have:

(ii)

Answer:

We know that

Answer:

(i)

(ii)

Answer:

(i)

(ii)

Answer:

(i) We have:

Hence, A is a symmetric matrix.

(ii) We have:

Hence, A is a skew-symmetric matrix.

Answer:

(i)

Hence, is a symmetric matrix.

(ii)

Hence, is a skew-symmetric matrix.

Answer:

The given matrix is

Answer:

(i)

Thus, is a symmetric matrix.

Thus, is a skew-symmetric matrix.

Representing A as the sum of P and Q:

(ii)

Thus, is a symmetric matrix.

Thus, is a skew-symmetric matrix.

Representing A as the sum of P and Q:

(iii)

Thus, is a symmetric matrix.

Thus, is a skew-symmetric matrix.

Representing A as the sum of P and Q:

(iv)

Thus, is a symmetric matrix.

Thus, is a skew-symmetric matrix.

Representing A as the sum of P and Q:

Answer:

The correct answer is A.

A and B are symmetric matrices, therefore, we have:

Thus, (AB − BA) is a skew-symmetric matrix.

Answer:

The correct answer is B.

Comparing the corresponding elements of the two matrices, we have:

Answer:

We know that A = IA

Answer:

We know that A = IA

Answer:

We know that A = IA

Answer:

We know that A = IA

Answer:

We know that A = IA

Answer:

We know that A = IA

Answer:

We know that A = AI

Answer:

We know that A = IA

Answer:

We know that A = IA

Answer:

We know that A = AI

Answer:

We know that A = AI

Answer:

We know that A = IA

Now, in the above equation, we can see all the zeros in the second row of the matrix on the L.H.S.

Therefore, A⁻¹ does not exist.

Answer:

We know that A = IA

Answer:

We know that A = IA

Applying, we have:

Now, in the above equation, we can see all the zeros in the first row of the matrix on the L.H.S.

Therefore, A⁻¹ does not exist.

Answer:

We know that A = IA

Answer:

We know that A = IA

Applying R₂ → R₂ + 3R₁ and R₃ → R₃ − 2R₁, we have:

Answer:

We know that A = IA

Applying, we have:

Answer:

Answer: D

We know that if A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is said to be the inverse of A. In this case, it is clear that A is the inverse of B.

Thus, matrices A and B will be inverses of each other only if AB = BA = I.

Answer:

It is given that

We shall prove the result by using the principle of mathematical induction.

For n = 1, we have:

Therefore, the result is true for n = 1.

Let the result be true for n = k.

That is,

Now, we prove that the result is true for n = k + 1.

Consider

From (1), we have:

Therefore, the result is true for n = k + 1.

Thus, by the principle of mathematical induction, we have:

Answer:

It is given that

We shall prove the result by using the principle of mathematical induction.

For n = 1, we have:

Therefore, the result is true for n = 1.

Let the result be true for n = k.

That is

Now, we prove that the result is true for n = k + 1.

Therefore, the result is true for n = k + 1.

Thus by the principle of mathematical induction, we have:

Answer:

It is given that

We shall prove the result by using the principle of mathematical induction.

For n = 1, we have:

Therefore, the result is true for n = 1.

Let the result be true for n = k.

That is,

Now, we prove that the result is true for n = k + 1.

Therefore, the result is true for n = k + 1.

Thus, by the principle of mathematical induction, we have:

Answer:

It is given that A and B are symmetric matrices. Therefore, we have:

Thus, (AB − BA) is a skew-symmetric matrix.

Answer:

We suppose that A is a symmetric matrix, then … (1)

Consider

Thus, if A is a symmetric matrix, thenis a symmetric matrix.

Now, we suppose that A is a skew-symmetric matrix.

Then,

Thus, if A is a skew-symmetric matrix, thenis a skew-symmetric matrix.

Hence, if A is a symmetric or skew-symmetric matrix, thenis a symmetric or skew-symmetric matrix accordingly.

Answer:

Now,

On comparing the corresponding elements, we have:

Answer:

We have:

∴4 + 4x = 0

⇒ x = −1

Thus, the required value of x is −1.

Answer:

Answer:

We have:

Answer:

(a) The unit sale prices of x, y, and z are respectively given as Rs 2.50, Rs 1.50, and Rs 1.00.

Consequently, the total revenue in market I can be represented in the form of a matrix as:

The total revenue in market II can be represented in the form of a matrix as:

Therefore, the total revenue in market I isRs 46000 and the same in market II isRs 53000.

(b) The unit cost prices of x, y, and z are respectively given as Rs 2.00, Rs 1.00, and 50 paise.

Consequently, the total cost prices of all the products in market I can be represented in the form of a matrix as:

Since the total revenue in market I isRs 46000, the gross profit in this marketis (Rs 46000 − Rs 31000) Rs 15000.

The total cost prices of all the products in market II can be represented in the form of a matrix as:

Since the total revenue in market II isRs 53000, the gross profit in this market is (Rs 53000 − Rs 36000) Rs 17000.

Answer:

It is given that:

The matrix given on the R.H.S. of the equation is a 2 × 3 matrix and the one given on the L.H.S. of the equation is a 2 × 3 matrix. Therefore, X has to be a 2 × 2 matrix.

Now, let

Therefore, we have:

Equating the corresponding elements of the two matrices, we have:

Thus, a = 1, b = 2, c = −2, d = 0

Hence, the required matrix X is

Answer:

A and B are square matrices of the same order such that AB = BA.

For n = 1, we have:

Therefore, the result is true for n = 1.

Let the result be true for n = k.

Now, we prove that the result is true for n = k + 1.

Therefore, the result is true for n = k + 1.

Thus, by the principle of mathematical induction, we have

Now, we prove that for all n ∈ N

For n = 1, we have:

Therefore, the result is true for n = 1.

Let the result be true for n = k.

Now, we prove that the result is true for n = k + 1.

Therefore, the result is true for n = k + 1.

Thus, by the principle of mathematical induction, we have, for all natural numbers.

Answer:

Answer: C

On comparing the corresponding elements, we have:

Answer:

Answer: B

If A is both symmetric and skew-symmetric matrix, then we should have

Therefore, A is a zero matrix.

Answer:

Answer: C