TOPICS

Exercise - 3.2

Matrices

**Question-1 :-**
Find each of the following:

(i) A + B

(ii) A – B

(iii) 3A – C

(iv) AB

(v) BA

(i)

(ii)

(iii)

(iv)

(v)

**Question-2 :-**
Compute the following:

(i)

(ii)

(iii)

(iv)

**Question-3 :-**
Compute the indicated products:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

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

**Question-5 :-**

**Question-6 :-**

**Question-7 :-**
Find X and Y, if

(i)

(ii)

**Question-8 :-**

**Question-9 :-**

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

**Question-11 :-**

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

**Question-13 :-**

**Question-14 :-**
Show that :

(i)

(ii)

**Question-15 :-**

**Question-16 :-**

**Question-17 :-**

**Question-18 :-**

**Question-19 :-**
A trust fund has ₹ 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 ₹ 30,000 among the two types of bonds.
If the trust fund must obtain an annual total interest of:

(a) ₹ 1800 (b) ₹ 2000

(i) Let ₹ x be invested in the first bond. Then, the sum of money invested in the second bond will be ₹ (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 ₹ 1800, we have: Thus, in order to obtain an annual total interest of ₹ 1800, the trust fund should invest ₹ 15000 in the first bond and the remaining ₹ 15000 in the second bond.

(ii) Let ₹ x be invested in the first bond. Then, the sum of money invested in the second bond will be ₹ (30000 − x). Therefore, in order to obtain an annual total interest of ₹ 2000, we have: Thus, in order to obtain an annual total interest of ₹ 2000, the trust fund should invest ₹ 5000 in the first bond and the remaining ₹ 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 ₹ 80, ₹ 60 and ₹ 40 each respectively.
Find the total amount the bookshop will receive from selling all the books using matrix algebra.

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 ₹ 80, ₹ 60, and ₹ 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 ₹ 20160 from the sale of all these books.

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

(A) k = 3, p = n (B) k is arbitrary, p = 2 (C) p is arbitrary, k = 3 (D) k = 2, p = 3

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 PY + WY will be defined. The correct answer is A.

**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.
If n = p, then the order of the matrix 7X – 5Z is:

(A) p × 2 (B) 2 × n (C) n × 3 (D) p × n

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. The correct answer is B.

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