# NCERT Solutions for Miscellaneous Exercise Chapter 3 Class 12 - Matrices

In this article, you will get NCERT Solutions for Class 12 Maths Chapter 3 Miscellaneous Exercise. These **miscellaneous exercise chapter 3** Class 12 consist of questions from all the different topics of this chapter. If you have done with all the exercises problems, you should be able to these miscellaneous exercise chapter 3 Class 12 problems. Miscellaneous exercises are considered to be difficult as compared to other exercises but students who have solved NCERT exercises on their own can solve these exercises easily. You may find difficulties while these problems, you can take help from NCERT Solutions for Class 12 Maths Chapter 3 Miscellaneous Exercise. Miscellaneous exercises are not very important for the board exams as over 90% of the questions in the board exams are not asked from Class 12 Maths chapter 3 miscellaneous exercise solutions. Miscellaneous exercises are very important for the students who are preparing for some competitive exams. Here you can check for NCERT solutions.

**Also, see**

Matrices Exercise 3.1

Matrices Exercise 3.2

Exercise 3.3

Matrices Exercise 3.4

**Matrices Miscellaneous Exercise:**

**Question:1 **Let , show that , where I is the identity matrix of order 2 and .

**Answer:**

Given :

To prove :

For n=1,

The result is true for n=1.

Let result be true for n=k,

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

Put the value of in above equation,

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

Thus, we have where ,.

**Question 2. **If then show that , .

**Answer:**

Given :

To prove:

For n=1, we have

Thus, the result is true for n=1.

Now, take n=k,

For, n=k+1,

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

Hence, we have , where .

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

**Answer:**

Given :

To prove:

For n=1, we have

Thus, result is true for n=1.

Now, take result is true for n=k,

For, n=k+1,

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

Hence, we have , where .

**Question 4. **If A and B are symmetric matrices, prove that is a skew symmetric matrix.

**Answer:**

If A, B are symmetric matrices then

and

we have,

Hence, we have

Thus,( AB-BA)' is skew symmetric.

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

**Answer:**

Let be a A is symmetric matrix , then

Consider,

Replace by

i.e.

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

Now, let A be a skew-symmetric matrix, then .

Replace by -,

i.e. .

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

Hence, the matrix B′AB is symmetric or skew-symmetric according to as A is symmetric or skew-symmetric.

**Question 6. **Find the values of *x*, *y*, *z* if the matrix satisfy the equation

**Answer:**

Thus equating the terms element wise

**Question 7. **For what values of *x: ?*

**Answer:**

*Thus, value of x is -1.*

**Question 8. **If , show that .

**Answer:**

To prove:

L.H.S :

Hence, we proved that

.

**Question 9. **Find *x,* if .

**Answer:**

thus the value of x is

**Question 10(a) **A manufacturer produces three products x, y, z which he sells in two markets.

Annual sales are indicated below:

** Market Products**

I 10,000 2,000 18,000

II 6,000 20,000 8,000

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.

**Answer:**

The unit sale prices of x, y and z are ` 2.50, ` 1.50 and ` 1.00, respectively.

The total revenue in the market I with the help of matrix algebra can be represented as :

The total revenue in market II with the help of matrix algebra can be represented as :

Hence, total revenue in the market I is 46000 and total revenue in market II is 53000.

**Question 10(b). **A manufacturer produces three products x, y, z which he sells in two markets.

Annual sales are indicated below:

** Market Products**

I 10,000 2,000 18,000

II 6,000 20,000 8,000

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

**Answer:**

The unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively.

The total cost price in market I with the help of matrix algebra can be represented as :

Total revenue in the market I is 46000 , gross profit in the market is

The total cost price in market II with the help of matrix algebra can be represented as :

Total revenue in market II is 53000, gross profit in the market is

**Question 11. **Find the matrix *X* so that

**Answer:**

The matrix given on R.H.S is matrix and on LH.S is matrix.Therefore, X has to be matrix.

Let X be

Taking,

Hence, we have

Matrix X is .

**Question 12. **If A and B are square matrices of the same order such that , then prove by induction that . Further, prove that for all .

**Answer:**

A and B are square matrices of the same order such that ,

To prove : ,

For n=1, we have

Thus, the result is true for n=1.

Let the result be true for n=k,then we have

Now, taking n=k+1 , we have

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

Hence, we have , .

To prove:

For n=1, we have

Thus, the result is true for n=1.

Let the result be true for n=k,then we have

Now, taking n=k+1 , we have

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

Hence, we have and for all .

**Question 13 **Choose the correct answer in the following questions:

** **If is such that

(A)

(B)

(C)

(D)

**Answer:**

Thus we obtained that

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

**Answer:**

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

and

Hence, A is a zero matrix.

Option B is correct.

**Question 15. **If A is square matrix such that , then is equal to

(A) *A*

(B) *I – A*

(C) *I*

(D) *3A*

**Answer:**

A is a square matrix such that

(Replace by )

Hence, we have

Option C is correct.

**More about ****NCERT Solutions for Class 12 Maths Chapter 3 Miscellaneous Exercise:-**

There are 3 miscellaneous solved examples are given in the NCERT textbook before themiscellaneous exercise which could be solved if you have conceptual clarity. You can solve the 12 long answer type questions and 3 multiple choice types questions given in the miscellaneous exercise of NCERT syllabus chapter 3 Class 12. Solving these problems will check your understanding of this chapter.

**Also Read| ****Matrices Class 12 Maths Chapter Notes**

**Benefits of NCERT for Class 12 Maths Chapter 3 ****Miscellaneous Exercise:-**

- Class 12 Maths chapter 3 miscellaneous exercise solutions are available at one place which are helpful for the students who are facing problems while solving them.
- First, try to solve previous exercises then you will be able to solve the miscellaneous exercise.
- Class 12 Maths chapter 3 miscellaneous exercise solutions are also helpful for the students who are preparing for the engineering entrance exams as some questions in these exams are directly asked from miscellaneous exercises.
- You can use Class 12 Maths chapter 3 miscellaneous solutions for quick revision before the exam.

**Also see-**

NCERT solutions for class 12 maths chapter 3

NCERT exemplar solutions class 12 maths chapter 3

**NCERT solutions of class 12 subject wise**

NCERT solutions for class 12 Maths

NCERT solutions for class 12 Physics

NCERT solutions for class 12 Chemistry

NCERT solutions for class 12 Biology

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