# NCERT Solutions for Exercise 3.2 Class 12 Maths Chapter 3 - Matrices

**NCERT solutions for class 12 maths chapter 3 exercise 3.2 **consist of questions related to operations on matrices like the addition of matrices, ** **multiplication of a matrix by a scalar, properties of matrix addition, and properties of scalar multiplication of a matrix. Topics such as properties of multiplication of matrices like associative, distributive, and existence of multiplicative identity are also covered in the NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.2. There are 22 questions given in exercise 3.2 class 12 maths solutions. You can take help from these class 12 maths ch 3 ex 3.2 solutions. You are advised to solve more problems to get conceptual clarity. You can solve some problems related to matrix multiplication from miscellaneous exercises also. You can also check for NCERT solutions.

**Also, see**

Matrices Exercise 3.1

Matrices Exercise 3.3

Matrices Exercise 3.4

Matrices Miscellaneous Exercise

**Matrices Exercise: 3.2**

**Question 1(i) **Let , ,

Find each of the following:

A + B

**Answer:**

(i) A + B

The addition of matrix can be done as follows

**Question 1(ii) **Let , ,

Find each of the following:

A - B

**Answer:**

(ii) A - B

**Question 1(iii) **Let , ,

Find each of the following:

3A - C

**Answer:**

(iii) 3A - C

First multiply each element of A with 3 and then subtract C

**Question 1(iv)**Let , ,

Find each of the following:

AB

**Answer:**

(iv) AB

**Question 1(v) **Let , ,

Find each of the following:

BA

**Answer:**

The multiplication is performed as follows

,

**Question 2(i). **Compute the following:

**Answer:**

(i)

**Question 2(ii). **Compute the following:

**Answer:**

(ii) The addition operation can be performed as follows

**Question 2(iii). **Compute the following:

**Answer:**

(iii) The addition of given three by three matrix is performed as follows

**Question 2(iv). **Compute the following:

**Answer:**

(iv) the addition is done as follows

since

**Question 3(i). **Compute the indicated products.

**Answer:**

(i) The multiplication is performed as follows

**Question 3(ii). **Compute the indicated products.

**Answer:**

(ii) the multiplication can be performed as follows

**Question 3(iii). **Compute the indicated products.

**Answer:**

(iii) The multiplication can be performed as follows

**Question 3(iv). **Compute the indicated products.

**Answer:**

(iv) The multiplication is performed as follows

**Question 3(v). **Compute the indicated products.

**Answer:**

(v) The product can be computed as follows

**Question 3(vi). **Compute the indicated products.

**Answer:**

(vi) The given product can be computed as follows

**Question 4. **If , and , then compute (A+B) and (B-C). Also verify that A + (B - C) = (A + B) - C

**Answer:**

, and

Now, to prove A + (B - C) = (A + B) - C

(Puting value of from above)

Hence, we can see L.H.S = R.H.S =

**Question 5. **If and , then compute 3A - 5B

**Answer:**

and

**Question 6. **Simplify .

**Answer:**

The simplification is explained in the following step

the final answer is an identity matrix of order 2

**Question 7(i). **Find X and Y, if

and

**Answer:**

(i) The given matrices are

and

Adding equation 1 and 2, we get

Putting the value of X in equation 1, we get

**Question 7(ii). **Find X and Y, if

and

**Answer:**

(ii) and

Multiply equation 1 by 3 and equation 2 by 2 and subtract them,

Putting value of Y in equation 1 , we get

**Question 8. **Find X, if and

**Answer:**

Substituting the value of Y in the above equation

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

**Answer:**

Now equating LHS and RHS we can write the following equations

**Question 10. **Solve the equation for x, y, z and t, if

**Answer:**

Multiplying with constant terms and rearranging we can rewrite the matrix as

Dividing by 2 on both sides

**Question 11. **If , find the values of x and y.

**Answer:**

Adding both the matrix in LHS and rewriting

Adding equation 1 and 2, we get

Put the value of x in equation 2, we have

**Question 12. **Given , find the values of x, y, z and w.

**Answer:**

If two matrices are equal than corresponding elements are also equal.

Thus, we have

Put the value of x

Hence, we have

**Question 13. **If , show that .

**Answer:**

To prove :

Hence, we have L.H.S. = R.H.S i.e. .

**Question 14(i).** Show that

**Answer:**

To prove:

Hence, the right-hand side not equal to the left-hand side, that is

**Question 14(ii). **Show that

**Answer:**

To prove the following multiplication of three by three matrices are not equal

Hence, i.e. .

**Question 15. **Find, if

**Answer:**

First, we will find ou the value of the square of matrix A

**Question 16. **If prove that .

**Answer:**

First, find the square of matrix A and then multiply it with A to get the cube of matrix A

L.H.S :

Hence, L.H.S = R.H.S

i.e..

**Question 17. **If and , find *k* so that .

**Answer:**

We have,

Hence, the value of k is 1.

**Question 18. **If and I is the identity matrix of order 2, show that

**Answer:**

To prove :

L.H.S :

R.H.S :

Hence, we can see L.H.S = R.H.S

i.e. .

**Question 19(i). **A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

Rs. 1800

**Answer:**

Let Rs. x be invested in the first bond.

Money invested in second bond = Rs (3000-x)

The first bond pays 5% interest per year and the second bond pays 7% interest per year.

To obtain an annual total interest of Rs. 1800, we have

(simple interest for 1 year )

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

**Question 19(ii). **A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

Rs. 2000

**Answer:**

Let Rs. x be invested in the first bond.

Money invested in second bond = Rs (3000-x)

The first bond pays 5% interest per year and the second bond pays 7% interest per year.

To obtain an annual total interest of Rs. 1800, we have

(simple interest for 1 year )

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

**Question 20. **The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

**Answer:**

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

Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively.

The total amount the bookshop will receive from selling all the books:

The total amount the bookshop will receive from selling all the books is 20160.

**Question 21 **Assume X, Y, Z, W and P are matrices of order *2 × n, 3 × k, 2 × p, n × 3* and* p × k*, respectively. Choose the correct answer in Exercises 21 and 22.

**Q21. **The restriction on n, k and p so that PY + WY will be defined are:

(A)

(B) k is arbitrary,

(C) p is arbitrary,

(D)

**Answer:**

P and Y are of order and respectivly.

PY will be defined only if k=3, i.e. order of PY is .

W and Y are of order and respectivly.

WY is defined because the number of columns of W is equal to the number of rows of Y which is 3, i.e. the order of WY is

Matrices PY and WY can only be added if they both have same order i.e = implies p=n.

Thus, are restrictions on n, k and p so that PY + WY will be defined.

Option (A) is correct.

**Question 22** Assume X, Y, Z, W and P are matrices of order *2 × n, 3 × k, 2 × p, n × 3* and* p × k*,

respectively. Choose the correct answer in Exercises 21 and 22.** **If *n = p*, then the order of the matrix is:

(A) *p × 2*

(B) *2 × n*

(C)* n × 3*

(D) *p × n*

**Answer:**

X has of order .

7X also has of order .

Z has of order .

5Z also has of order .

Mtarices 7X and 5Z can only be subtracted if they both have same order i.e = and it is given that p=n.

We can say that both matrices have order of .

Thus, order of is .

Option (B) is correct.

**More about NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.2:-**

There are 20 long answer type questions and 2 multiple-type questions are given in the NCERT Solutions for **Class 12 Maths Chapter 3 Exercise 3.2**. You should try to solve all of them on your own. Also, there are 14 solved examples given before the **NCERT textbook** exercise 3.2 Class 12 Maths. Solving these examples will help you to grasp the concepts and solve textbook questions very easily. These Class 12th maths chapter 3 exercise 3.2 examples are very descriptive with help some important definitions. There are some theorems given in the textbook. Sometimes prove of these theorems is asked in the CBSE board exams. You should look into them also.

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

**Benefits of NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.2:-**

- NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.2 are very helpful in solving NCERT problems.
- Class 12 maths chapter 3 exercise 3.2 solutions are designed the basis of guidelines given by the CBSE on which you can rely upon.
- Only Knowing the answer is enough to get good marks but you should know how to write answers in the board exams in order to get good marks.
- Exercise 3.2 Chapter 3 Maths Solutions are prepared by experts who know how best to write in the board exams.
- You can use these class 12 maths chapter 3 exercise 3.2 solutions. for reference.

**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

**Subject wise NCERT Exampler solutions**

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