Advertisements
Advertisements
प्रश्न
If `P(x)=[[cos x,sin x],[-sin x,cos x]],` then show that `P(x),P(y)=P(x+y)=P(y)P(x).`
उत्तर
Given : \[P(x) = \begin{bmatrix}\ cosx & \ sinx \\ - \ sinx & \ cosx\end{bmatrix}\]
Then ,
\[P(y) = \begin{bmatrix}\ cosy & \ siny \\ - \ siny & \ cosy\end{bmatrix}\]
Now,
\[P(x) P(y) = \begin{bmatrix}\ cosx & \ sinx \\ - \ sinx & \ cosx\end{bmatrix}\begin{bmatrix}\ cosy & \ siny \\ - \ siny & \ cosy\end{bmatrix}\]
\[ = \begin{bmatrix}\ cosx\ cosy - \ sinx\ siny & \ cosx\ siny + \ sinx\ cosy \\ - \ sinx\ cosy - \ cosx\ siny & - \ sinx\ siny + \ cosx\ cosy\end{bmatrix}\]
\[ = \begin{bmatrix}\cos\left( x + y \right) & \sin\left( x + y \right) \\ - \sin\left( x + y \right) & \cos\left( x + y \right)\end{bmatrix} . . . (1)\]
Also,
\[P(x + y) = \begin{bmatrix}\cos\left( x + y \right) & \sin\left( x + y \right) \\ - \sin\left( x + y \right) & \cos\left( x + y \right)\end{bmatrix} . . . (2)\]
Now,
\[P(y) P(x) = \begin{bmatrix}\ cosy & \ siny \\ - \ siny & \ cosy\end{bmatrix}\begin{bmatrix}\ cosx & \ sinx \\ - \ sinx & \ cosx\end{bmatrix}\]
\[ = \begin{bmatrix}\ cosy\ cosx - \ siny\ sinx & \ cosy\ sinx + \ siny\ cosx \\ - \ siny \ cosx - \ cosy\ sinx & - \ siny\ sinx + \ cosy\ cosx\end{bmatrix}\]
\[ = \begin{bmatrix}\cos\left( x + y \right) & \sin\left( x + y \right) \\ - \sin\left( x + y \right) & \cos\left( x + y \right)\end{bmatrix} . . . (3)\]
from (1),(2) and (3) ,we get
`p(x) p(y)=p(x+y)=p(y) p(x)`
APPEARS IN
संबंधित प्रश्न
Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`
Find BA
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 2,000.
If A =`[[2 -3 -5],[-1 4 5],[1 -3 -4]]` and B =`[[2 -2 -4],[-1 3 4],[1 2 -3]]`
, show that AB = A and BA = B.
For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:
`A = [[1 -1],[0 2]] B= [[-1 0],[2 1]]`and `C= [[0 1],[1 -1]]`
Compute the elements a43 and a22 of the matrix:`A=[[0 1 0],[2 0 2],[0 3 2],[4 0 4]]` `[[2 -1],[-3 2],[4 3]] [[0 1 -1 2 -2],[3 -3 4 -4 0]]`
\[A = \begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}\] , Show that A2 = A.
If \[A = \begin{bmatrix}4 & - 1 & - 4 \\ 3 & 0 & - 4 \\ 3 & - 1 & - 3\end{bmatrix}\] , Show that A2 = I3.
If
Show that the matrix \[A = \begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix}\] is root of the equation A2 − 12A − I = O
If
Find the value of x for which the matrix product`[[2 0 7],[0 1 0],[1 -2 1]]` `[[-x 14x 7x],[0 1 0],[x -4x -2x]]`equal an identity matrix.
Solve the matrix equations:
`[1 2 1] [[1,2,0],[2,0,1],[1,0 ,2]][[0],[2],[x]]=0`
Solve the matrix equations:
`[[],[x-5-1],[]][[1,0,2],[0,2,1],[2,0,3]] [[x],[4],[1]]=0`
If , then show that A is a root of the polynomial f (x) = x3 − 6x2 + 7x + 2.
`A=[[3,-5],[-4,2]]` then find A2 − 5A − 14I. Hence, obtain A3
If `A=[[1,1],[0,1]] ,` Prove that `A=[[1,n],[0,1]]` for all positive integers n.
Give examples of matrices
A, B and C such that AB = AC but B ≠ C, A ≠ 0.
If A and B are square matrices of the same order, explain, why in general
(A + B) (A − B) ≠ A2 − B2
Three shopkeepers A, B and C go to a store to buy stationary. A purchases 12 dozen notebooks, 5 dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils. C purchases 11 dozen notebooks, 13 dozen pens and 8 dozen pencils. A notebook costs 40 paise, a pen costs Rs. 1.25 and a pencil costs 35 paise. Use matrix multiplication to calculate each individual's bill.
The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are Rs. 8.30, Rs. 3.45 and Rs. 4.50 each respectively. Find the total amount the store will receive from selling all the items.
There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B. The recommend daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrix. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the two families. What awareness can you create among people about the planned diet from this question?
For any square matrix write whether AAT is symmetric or skew-symmetric.
If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, find the order of the matrix of AB.
Write a 2 × 2 matrix which is both symmetric and skew-symmetric.
If `[2 1 3]([-1,0,-1],[-1,1,0],[0,1,1])([1],[0],[-1])=A` , then write the order of matrix A.
If \[A = \begin{bmatrix}1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} and B = \begin{bmatrix}1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\] and AB = I3, then x + y equals
If \[\begin{bmatrix}2x + y & 4x \\ 5x - 7 & 4x\end{bmatrix} = \begin{bmatrix}7 & 7y - 13 \\ y & x + 6\end{bmatrix}\]
A matrix which is not a square matrix is called a ______ matrix.
If A and B are square matrices of the same order, then [k (A – B)]′ = ______.
If A and B are two square matrices of the same order, then AB = BA.
Three schools DPS, CVC, and KVS decided to organize a fair for collecting money for helping the flood victims. They sold handmade fans, mats, and plates from recycled material at a cost of Rs. 25, Rs.100, and Rs. 50 each respectively. The numbers of articles sold are given as
| School/Article | DPS | CVC | KVS |
| Handmade/fans | 40 | 25 | 35 |
| Mats | 50 | 40 | 50 |
| Plates | 20 | 30 | 40 |
Based on the information given above, answer the following questions:
- What is the total money (in Rupees) collected by the school DPS?
Three schools DPS, CVC, and KVS decided to organize a fair for collecting money for helping the flood victims. They sold handmade fans, mats, and plates from recycled material at a cost of Rs. 25, Rs.100, and Rs. 50 each respectively. The numbers of articles sold are given as
| School/Article | DPS | CVC | KVS |
| Handmade/fans | 40 | 25 | 35 |
| Mats | 50 | 40 | 50 |
| Plates | 20 | 30 | 40 |
Based on the information given above, answer the following questions:
- What is the total amount of money (in Rs.) collected by schools CVC and KVS?
If A = `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)]`, B = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]` then find AB and use it to solve the following system of equations:
x – 2y = 3
2x – y – z = 2
–2y + z = 3
