Advertisements
Advertisements
प्रश्न
If `A=[[i,0],[0,i ]]` , n ∈ N, then A4n equals
पर्याय
`[[0,I],[I,0]]`
`[[0,0],[0,0]]`
`[[1,0],[0,1]]`
`[[0,I],[I,0]]`
उत्तर
`[[1 0],[0 1]]`
\[Here, \]
\[A = \begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix}\begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix} = \begin{bmatrix}i^2 & 0 \\ 0 & i^2\end{bmatrix}\]
\[ \Rightarrow A^3 = A^2 . A = \begin{bmatrix}i^2 & 0 \\ 0 & i^2\end{bmatrix}\begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix} = \begin{bmatrix}- i & 0 \\ 0 & - i\end{bmatrix}\]
\[ \Rightarrow A^4 = A^3 . A = \begin{bmatrix}- i & 0 \\ 0 & - i\end{bmatrix}\begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
So,
APPEARS IN
संबंधित प्रश्न
Compute the indicated product.
`[(1),(2),(3)] [2,3,4]`
Compute the indicated products
`[(2,3,4),(3,4,5),(4,5,6)][(1,-3,5),(0,2,4), (3,0,5)]`
Compute the indicated products:
`[[a b],[-b a]][[a -b],[b a]]`
Compute the products AB and BA whichever exists in each of the following cases:
[a, b]`[[c],[d]]`+ [a, b, c, d] `[[a],[b],[c],[d]]`
Evaluate the following:
`[[],[1 2 3],[]]` `[[1 0 2],[2 0 1],[0 1 2]]` `[[2],[4],[6]]`
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 associativity of matrix multiplication i.e. (AB) C = A(BC):
`A =-[[1 2 0],[-1 0 1]]`,`B=[[1 0],[-1 2],[0 3]]` and C= `[[1],[-1]]`
Show that the matrix \[A = \begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix}\] is root of the equation A2 − 12A − I = O
Solve the matrix equations:
`[x1][[1,0],[-2,-3]][[x],[5]]=0`
Solve the matrix equations:
[2x 3] `[[1 2],[-3 0]] , [[x],[8]]=0`
If `A= [[1,2,0],[3,-4,5],[0,-1,3]]` compute A2 − 4A + 3I3.
Find the matrix A such that `=[[1,2,3],[4,5,6]]=[[-7,-8,-9],[2,4,6],[11,10,9]]`
If\[A = \begin{bmatrix}a & b \\ 0 & 1\end{bmatrix}\], prove that\[A^n = \begin{bmatrix}a^n & b( a^n - 1)/a - 1 \\ 0 & 1\end{bmatrix}\] for every positive integer n .
Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2 B2? Give reasons.
If A and B are square matrices of the same order, explain, why in general
(A + B) (A − B) ≠ A2 − B2
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.
A trust fund has Rs 30000 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 30000 among the two types of bonds. If the trust fund must obtain an annual total interest of(ii) Rs 2000
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
\[A = \begin{bmatrix}140 \\ 200 \\ 150\end{bmatrix}\begin{array}Telephone \\ House calls \\ Letters\end{array}\]
The number of contacts of each type made in two cities X and Y is given in the matrix B as
\[\begin{array}Telephone & House calls & Letters\end{array}\]
\[B = \begin{bmatrix}1000 & 500 & 5000 \\ 3000 & 1000 & 10000\end{bmatrix}\begin{array} \\City X \\ City Y\end{array}\]
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?
Let `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that
(A + B)T = AT + BT
Let `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that
(A − B)T = AT − BT
Let `A= [[1,-1,0],[2,1,3],[1,2,1]]` And `B=[[1,2,3],[2,1,3],[0,1,1]]` Find `A^T,B^T` and verify that (A + B)T = AT + BT
If \[A = \begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\] satisfies A4 = λA, then write the value of λ.
If A = [aij] is a square matrix such that aij = i2 − j2, then write whether A is symmetric or skew-symmetric.
If \[\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\begin{bmatrix}3 & 1 \\ 2 & 5\end{bmatrix} = \begin{bmatrix}7 & 11 \\ k & 23\end{bmatrix}\] ,then write the value of k.
Write a 2 × 2 matrix which is both symmetric and skew-symmetric.
If AB = A and BA = B, where A and B are square matrices, then
If A is a matrix of order m × n and B is a matrix such that ABT and BTA are both defined, then the order of matrix B is
Disclaimer: option (a) and (d) both are the same.
If A = `[(2, -1, 3),(-4, 5, 1)]` and B = `[(2, 3),(4, -2),(1, 5)]`, then ______.
If A = `[(3, 5)]`, B = `[(7, 3)]`, then find a non-zero matrix C such that AC = BC.
Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: A(BC) = (AB)C
Prove by Mathematical Induction that (A′)n = (An)′, where n ∈ N for any square matrix A.
A square matrix where every element is unity is called an identity matrix.
If A and B are two square matrices of the same order, then AB = BA.
If A, B and C are square matrices of same order, then AB = AC always implies that B = C
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?
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:
- How many articles (in total) are sold by three schools?
If A = `[(a, b),(b, a)]` and A2 = `[(α, β),(β, α)]`, then ______.
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
