Class 12 NCERT Solutions- Mathematics Part I - Chapter 3 Matrices - Miscellaneous Exercise on Chapter 3

Chapter 3 of the Class 12 NCERT Mathematics textbook, titled "Matrices," covers fundamental concepts related to matrices, including operations such as addition, multiplication, and finding determinants and inverses. The Miscellaneous Exercise in this chapter provides a range of problems that integrate these concepts, helping students apply their understanding of matrices in various contexts and deepen their problem-solving skills.

NCERT Solutions for Class 12 - Mathematics Part I - Chapter 3 Matrices - Miscellaneous Exercise

This section provides detailed solutions for the Miscellaneous Exercise from Chapter 3 of the Class 12 NCERT Mathematics textbook. The exercise includes problems on matrix operations, determinants, and inverses, designed to challenge students and reinforce their comprehension of matrix theory. The solutions offer step-by-step explanations to ensure a thorough understanding of each concept.

Question 1: Let , show that (aI + bA)ⁿ = aⁿ I + na^{n – 1} bA, where I is the identity matrix of order 2 and n ∈ N.

Solution:

Using mathematical induction,

Let's check for n=1

(aI + bA)ⁿ = (aI + bA)¹ = (aI + bA)

aⁿI + na^{n – 1} bA = aI + 1a^{1 – 1} bA = (aI + bA)

It is true for P(1)

Now take n=k

(aI + bA)^k = a^kI + ka^{k – 1} bA .....................(1)

Let's check whether, its true for n = k+1

(aI + bA)^k+1 = (aI + bA)^k (aI + bA)

= (a^kI + ka^{k – 1} bA) (aI + bA)

= a^k+1I×I + ka^k bAI + a^k bAI + ka^k-1 b²AA

AA = 

= a^k+1I×I + ka^k bAI + a^k bAI + 0

= a^k+1I + (k+1)a^k+1-1 bA

= P(k+1)

Hence, P(n) is true.

Question 2: If , prove that 

Solution:

Using mathematical induction,

: Let's check for n=1

It is true for P(1)

: Now take n=k

Let's check whether, its true for n = k+1

= P(k+1)

Hence, P(n) is true.

Question 3: If , prove that  ,where n is any positive integer.

Solution:

Using mathematical induction,

Step 1: Let's check for n=1

It is true for P(1)

Step 2: Now take n=k

Step 3: Let's check whether, its true for n = k+1

= P(k+1)

Hence, P(n) is true.

Question 4. If A and B are symmetric matrices, prove that AB – BA is a skew-symmetric matrix.

Solution:

As, it is mentioned that A and B are symmetric matrices,

A' = A and B' = B

(AB – BA)' = (AB)' - (BA)'  (using, (A-B)' = A' - B')

= B'A' - A'B'                     (using, (AB)' = B'A')

= BA - AB

(AB – BA)' = - (AB - BA)

Hence, AB – BA is a skew symmetric matrix

Question 5. Show that the matrix B′AB is symmetric or skew-symmetric according as A is symmetric or skew-symmetric.

Solution:

Let's take A as symmetric matrix

A' = A

Then,

(B′AB)' = {B'(AB)}'

= (AB)' (B')'            (using, (AB)' = B'A')

= B'A' (B)              (using, (AB)' = B'A' and (B')' = B)

= B'A B

As, here (B′AB)' = B'A B. It is a symmetric matrix.

Let's take A as skew matrix

A' = -A

Then,

(B′AB)' = {B'(AB)}'

= (AB)' (B')'            (using, (AB)' = B'A')

= B'A' (B)              (using, (AB)' = B'A' and (B')' = B)

= B'(-A) B

= - B'A B

As, here (B′AB)' = -B'A B. It is a skew matrix.

Hence, we can conclude that B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Question 6. Find the values of x, y, z if the matrix  satisfy the equation A′A = I

Solution:

A'A = 

By evaluating the values, we have

2x² = 1

x = ± 

6y² = 1

y = ± 

3z² = 1

z = ± 

Question 7: For what values of x : 

Solution:

Question 8: If , show that A² – 5A + 7I = 0.

Solution:

A² – 5A + 7I = 

Hence proved!

Question 9: Find x, if 

Solution:

Question 10: A manufacturer produces three products x, y, z which he sells in two markets.

Annual sales are indicated below:

(a) If unit sale prices of x, y and z are ₹ 2.50, ₹ 1.50 and ₹ 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

Solution:

Total revenue in market I and II can be arranged from given data as follows:

After multiplication, we get

Hence, the total revenue in Market I and market II are ₹ 46,000 and ₹ 53,000 respectively.

(b) If the unit costs of the above three commodities are ₹ 2.00, ₹ 1.00 and 50 paise respectively. Find the gross profit.

Solution:

Total cost prices of all the products in market I and market II can be arranged from given data as follows:

After multiplication, we get

As, Profit earned = Total revenue - Cost price

Profit earned 

Profit earned = 

Hence, profit earned in Market I and market II are ₹ 15,000 and ₹ 17,000 respectively. Which is equal to ₹ 32,000

Question 11. Find the matrix X so that 

Solution:

Here, the RHS is a 2×3 matrix and LHS is 2×3. So, X will be 2×2 matrix.

Let's take X as,

Now solving the matrix, we have

Equating each of them, we get

p+4q = -7 ...........(1)

2p+5q = -8 .............(2)

3p + 6q = -9

r + 4s = 2 ............(3)

2r + 5s = 4 ...............(5)

3r + 6s = 6

Solving (1) and (2), we get

p = 1 and q = -2

Solving (3) and (4), we get

r = 2 and s = 0

Hence, matrix X is 

Question 12: If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABⁿ = BⁿA. Further, prove that (AB)ⁿ = AⁿBⁿ for all n ∈ N.

Solution:

Using mathematical induction,

Let's check for n=1

ABⁿ = AB¹ = AB

BⁿA = B¹A = BA

It is true for P(1)

Now take n=k

AB^k = B^kA

: Let's check whether, its true for n = k+1

AB^(k+1) = AB^kB

= B^kAB

= B^k+1 A

= P(k+1)

Hence, P(n) is true.

Now, for (AB)ⁿ = AⁿBⁿ

Using mathematical induction,

: Let's check for n=1

(AB)¹ = AB

B¹A¹ = BA

It is true for P(1)

: Now take n=k

(AB)^k = A^kB^k

: Let's check whether, its true for n = k+1

(AB)^(k+1) = (AB)^k(AB)

= A^kB^k AB

= A^k+1 B^k+1

= (AB)^k+1

= P(k+1)

Hence, P(n) is true.

Choose the correct answer in the following questions: 

Question 13: If  is such that A² = I, then

(A) 1 + α² + βγ = 0 

(B) 1 – α² + βγ = 0

(C) 1 – α² – βγ = 0 

(D) 1 + α² – βγ = 0

Solution:

As, A² = I

α² + βγ = 1

1 - α² - βγ = 0

Hence, Option (C) is correct.

Question 14. If the matrix A is both symmetric and skew symmetric, then

(A) A is a diagonal matrix 

(B) A is a zero matrix

(C) A is a square matrix 

(D) None of these

Solution:

If the matrix A is both symmetric and skew symmetric, then

A = A'

and A = -A

Only zero matrix satisfies both the conditions.

Hence, Option (B) is correct.

Question 15. If A is square matrix such that A² = A, then (I + A)³ – 7 A is equal to

(A) A 

(B) I – A 

(C) I 

(D) 3A

Solution:

(I + A)³ – 7 A = I³ + A³ + 3A^2 + 3AI^2 - 7A

= I3 + A3 + 3A² + 3A - 7A

= I + A³ + 3A² - 4A

As, A² = A

A3 = A²A = AA = A

So, I + A3 + 3A2 - 4A = I + A + 3A - 4A = I

Hence, Option (C) is correct.

Summary

Chapter 3 of the Class 12 NCERT Mathematics textbook, "Matrices," explores various matrix operations and concepts. The Miscellaneous Exercise provides diverse problems on matrix addition, multiplication, determinants, and inverses. It challenges students to apply these concepts and reinforces their understanding of matrix theory through detailed problem-solving.

Related Articles:

• Class 12 NCERT Solutions- Mathematics Part II - Chapter 3 Matrices - Miscellaneous Exercise on Chapter 3
• Class 12 NCERT Solutions- Mathematics Part I - Chapter 2 - Miscellaneous Exercise on Chapter 2