Graded Assignment: Week 1

Problem 1 (1 point)

Suppose there are two types of oranges and two types of bananas available in the market. Suppose $1 \text{ kg}$ of each type of orange costs Rs. $50$ and $1 \text{ kg}$ of each type of banana costs Rs. $40$ . Gargi bought $x \text{ kg}$ of the first type of each fruit, orange and banana, and $y \text{ kg}$ of the second type of each fruit, orange and banana. She paid Rs. $250$ for oranges and Rs. $200$ for bananas.

Which of the following options are correct with respect to the given information?

The matrix representation to find $x$ and $y$ can be $\begin{bmatrix} 50 & 50 \\ 40 & 40 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 250 \\ 200 \end{bmatrix}$

The matrix representation to find $x$ and $y$ can be $\begin{bmatrix} 50 & 40 \\ 50 & 40 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 250 \\ 200 \end{bmatrix}$

The matrix representation to find $x$ and $y$ can be $\begin{bmatrix} 40 & 40 \\ 50 & 50 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 200 \\ 250 \end{bmatrix}$

x

can be 2 and

y

can be 3.

There are infinitely many real values possible for

x

and

y

.

There are only finitely many real values possible for

x

and

y

.

There are only finitely many natural numbers possible for

x

and

y

.

Problem 2

Suppose $det(4A) = n \times det(A)$ for any $5 \times 5$ real matrix $A$ . What is the value of $n$ ?

Problem 3

Let $A$ be a square matrix such that $A^2 = A$ . If $(I + A)^3 - 17A = I + mA$ , then find the value of $m$ .

Problem 4

If $A = \begin{bmatrix} 20 & 30 & 40 \\ 10 & 20 & 30 \\ 4 & 5 & 6 \end{bmatrix}$ , then what will be the determinant of $A$ ?

Problem 5

Let $A$ be a square matrix of order 3 and $B$ be a matrix that is obtained by adding 9 times the first row of $A$ to the third row of $A$ and adding 4 times the second row of $A$ to the first row of $A$ . If $det(A) = 4$ , then find out the value of $det(10A^2B^{-1})$ .

Problem 6

If $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$ , then what will be the value of the sum of the diagonal elements of $A^6$ ?

Problem 7

Let $A = [a_{ij}]$ be a square matrix of order 3, where $a_{ij} = 5i + 4j$ . Find $det(A)$ .

Problem 8 (1 point)

The scores of a student in mathematics, physics and chemistry in her class-12 board exams are $m$ , $p$ and $c$ respectively, where each score is out of 100. She has applied for three engineering streams in a college. Each stream assigns different weights to these three subjects to calculate her final score, which is again out of 100.

For example, the weight given to mathematics, physics and chemistry could be 0.2, 0.7 and 0.1 respectively by a stream. These weights are then multiplied with the corresponding scores to get the final score. Concretely, if the student has scored 85, 78, 40 in the three subjects, her final score for this stream is: $0.2 \times 85 + 0.7 \times 78 + 0.1 \times 40 = 75.6$

This is called a weighted average. Note that the weights always sum to 1. Now, the weights assigned by the three streams for mathematics, physics and chemistry, in this order, are given below:

Stream-1: 0.2, 0.7, 0.1 Stream-2: 0.5, 0.3, 0.2 Stream-3: 0.1, 0.4, 0.5

The final score of the student in stream-1 is 81. It is 83 in stream-2 and 76 in stream-3. We wish to find the student’s marks in the three subjects.

This is framed as a system of linear equations. Select all true options concerning the coefficient matrix if the vector of unknowns is given as $\begin{bmatrix} m \\ p \\ c \end{bmatrix}$ . Assume that the first equation corresponds to stream-1, second to stream-2 and last to stream-3.

The first row is

0.2, 0.7, 0.1

The last row is

0.1, 0.4, 0.5

The first column is

0.2, 0.7, 0.1

The middle column is

0.5, 0.3, 0.2