In a city there are two factories A and B. Each factory produces sports clothes for boys and girls. There are three types of clothes produced in both the factories, type I, II and III. For boys the number of units of types I, II and III respectively are 80, 70'and 65 in factory A and 85, 65 and 72 are in factory B. For girls the number of units of types I, II and III respectively are 80, 75, 90 in factory A and 50, 55, 80 are in factory B.
 
Based on the above information, answer the following questions.
(i) If P represents the matrix of number of units of each type produced by factory A for both boys and girls, then P is given by

(ii) If Q represents the matrix of number of units of each type produced by factory B for both boys and girls, then Q is given by
 
(iii) The total- production of sports clothes of each type for boys is given by the matrix
 
(iv) The total production of sports clothes of each type for girls is given by the matrix

(v) Let R be a 3 x 2 matrix that represent the total production of sports clothes of each type for boys and girls, then transpose of R is(iv) The total production of sports clothes of each type for girls is given by the matrix

Three car dealers, say A, Band C, deals in three types of cars, namely Hatchback cars, Sedan cars, SUV cars. The sales figure of 2019 and 2020 showed that dealer A sold 120 Hatchback, 50 Sedan, 10 SUV cars in 2019 and 300 Hatchback, 150 Sedan, 20 SUV cars in 2020; dealer B sold 100 Hatchback, 30 Sedan,S SUV cars in 2019 and 200 Hatchback, 50 Sedan, 6 SUV cars in 2020; dealer C sold 90 Hatchback, 40 Sedan, 2 SUV cars in 2019 and 100 Hatchback, 60 Sedan,S SUV cars in 2020.
 
Based on the above information, answer the following questions.
(i) The matrix summarizing sales data of 2019 is
 
(ii) The matrix summarizing sales data of 2020 is
 
 (iii) The total number of cars sold in two given years, by each dealer, is given by the matrix
 
(iv) The increase in sales from 2019 to 2020 is given by the matrix
 
(v) If each dealer receive profit of Rs. 50000 on sale of a Hatchback, Rs. 100000 on sale of a Sedan and Rs. 200000 on sale of a SUV (v) then amount of profit received in the year 2020 by each dealer is given by the matrix.

 

Three schools A, Band C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs. 25, Rs.100 and Rs.50 each. The number of articles sold by school A, B, C are given below.

Based on above information, answer the following questions.
(i) If P be a 3 x 3 matrix represent the sale of handmade fans, mats and plates by three schools A, Band C, then
 
(ii) If Q be a 3 x 1 matrix represent the sale prices (in Rs) of given products per unit, then

(iii) The funds collected by school A by selling the given articles is

(iv) The funds collected by school B by selling the given articles is

(v) The total funds collected for the required purpose is

A manufacturer produces three types of bolts, x, y and z which he sells in two markets. Annual sales (in Rs) are indicated below:

If unit sales prices of x, y and z are Rs.2.50, Rs.1.50 and Rs.1.00 respectively, then answer the following questions using the concept of matrices.
(i) Find the total revenue collected from the Market-I.

(ii) Find the total revenue collected from the Market-II.

(iii) If the unit costs of the above three commodities are Rs.2.00, Rs.1.00 and 50 paise respectively, then find the gross profit from both the markets.

(iv) If matrix \(4=\left[a_{i j}\right]_{2 \times 2}\) , where \(a_{i j}=1, \text { if } i \neq j\) , and \(a_{i j}=0 \text { if } i=j\) , then A^{2 i}s equal to

(v) If A and B are matrices of same order, then (AB' - BA') is a

If \(A=\left[a_{i j}\right]_{m \times n} \text { and } B=\left[b_{i j}\right]_{m \times n}\) are two matrices, then \(A \pm B\) is of order m x n and is defined as \((A \pm B)_{i j}=a_{i j} \pm b_{i j}\), where i = 1,2, , m and  j = 1,2, ..., n
If \(A=\left[a_{i j}\right]_{m \times n} \text { and } B=\left[b_{j k}\right]_{n \times p}\) are two matrices, then AB is of order m x p and is defined as \((A B)_{i k}=\sum_{r=1}^{n} a_{i r} b_{r k}=a_{i 1} b_{1 k}+a_{i 2} b_{2 k}+\ldots . .+a_{i n} b_{n k}\) 
Consider \(A=\left[\begin{array}{cc} 2 & -1 \\ 3 & 4 \end{array}\right], B=\left[\begin{array}{ll} 5 & 2 \\ 7 & 4 \end{array}\right], C=\left[\begin{array}{ll} 2 & 5 \\ 3 & 8 \end{array}\right] \text { and } D=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\) 
Using the concept of matrices answer the following questions. 
(i) Find the product AB. 

(ii) If A and B are any other two matrices such that AB exists, then

(iii) Find the values of a and c in the matrix D such than CD - AB = 0.

(iv) Find the values of band d in the matrix D such that CD - AB = 0.

(v) Find B + D.