Question 27:

If $A = [\begin{matrix} 3 & - 2 \\ 4 & - 2 \end{matrix}] and I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ , then prove that A² − A + 2I = O.

Answer:

Answer:

Shopkeepers	Notebooks In dozen	Pens In dozen	Pencils In dozen
A	12	5	6
B	10	6	7
C	11	3	8

Here,
Cost of notebooks per dozen =

(12 \times 40) paise

= Rs 4.80
Cost of pens per dozen =

Rs . (12 \times 1.25)

= Rs 15
Cost ofpPencils per dozen =

(12 \times 35) paise

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as

$A = [\begin{matrix} 140 \\ 200 \\ 150 \end{matrix}] \begin{matrix} Telephone \\ House calls \\ Letters \end{matrix}$

The number of contacts of each type made in two cities X and Y is given in the matrix B as

$\begin{matrix} Telephone & House calls & Letters \end{matrix} B = [\begin{matrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{matrix}] \begin{matrix} City X \\ City Y \end{matrix}$

Find the total amount spent by the party in the two cities.

What should one consider before casting his/her vote − party's promotional activity of their social activities?

Answer:

According to the question,

Let A be the matrix showing the cost per contact (in paisa).
$A = [\begin{matrix} 140 \\ 200 \\ 150 \end{matrix}] \begin{matrix} Telephone \\ House calls \\ Letters \end{matrix}$

And, B be a matrix showing the number of contacts of each type made in two cities X and Y.
$\begin{matrix} Telephone & House calls & Letters \end{matrix} B = [\begin{matrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{matrix}] \begin{matrix} City X \\ City Y \end{matrix}$

Now, the total amount spent by the party in the two cities will be shown by BA.
$B A = [\begin{matrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{matrix}] [\begin{matrix} 140 \\ 200 \\ 150 \end{matrix}] = [\begin{matrix} 140000 + 100000 + 750000 \\ 420000 + 200000 + 1500000 \end{matrix}] = [\begin{matrix} 990000 \\ 2120000 \end{matrix}]$

Hence, the total amount spent by the party in the two cities is
X: ₹9900
Y: ₹21200

One should consider social activities of a party before casting his/her vote.

Page No 5.48:

Question 78:

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹ 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Answer:

If I is the identity matrix and A is a square matrix such that A² = A, then what is the value of (I + A)² = 3A?

Answer:

Given: A is a square matrix, such that $A^{2} = A$ .

Here,
${(I + A)}^{2} - 3 A = (I + A) (I + A) - 3 A \Rightarrow {(I + A)}^{2} - 3 A = I \times I + I \times A + A \times I + A \times A - 3 A (using distributive property) \Rightarrow {(I + A)}^{2} - 3 A = I + A + A + A^{2} - 3 A (using I \times I = I and I A = A I = A) \Rightarrow {(I + A)}^{2} - 3 A = I + 2 A + A - 3 A (∵ A^{2} = A) \Rightarrow {(I + A)}^{2} - 3 A = I + 3$