Advertisements
Advertisements
प्रश्न
If A `= [(1,3),(3,4)]` and A2 − kA − 5I = 0, then the value of k is ______.
विकल्प
5
3
7
9
उत्तर
If A `= [(1,3),(3,4)]` and A2 - KA - 5I = 0, then K = 5.
Explanation:
Given,
`A = [(1,3),(3,4)]`
A2 = A.A
`=[(1,3),(3,4)] [(1,3),(3,4)]`
`= [((1)(1)+(3)(3),(1)(3)+(3)( 4)),((3)(1) + ( 4)(3),(3)(3) + ( 4)( 4))]`
`= [(1+9,3+12),(3+12,9+16)]`
`= [(10,15),(15,25)]`
Now, A2 − KA − 5I = 0 [Given]
`[(10,15),(15,25)]-k[(1,3),(3,4)]-5[(1,0),(0,1)]=[(0,0),(0,0)]`
`[(10-k-5,15-3k-0),(15-3k-0,25-4k-5)]=[(0,0),(0,0)]`
`[(5-k,15-3k),(15-3k,20-4k)]=[(0,0),(0,0)]`
On comparing
5 − k = 0
∴ k = 5
OR 15 - 3k = 0
∴ k = 5
OR 20 - 4k = 0
Hence, k = 5
APPEARS IN
संबंधित प्रश्न
Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`
Find AB
Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`
Find BA
Compute the indicated product.
`[(1),(2),(3)] [2,3,4]`
If [1 −1 x] `[[0 1 -1],[2 1 3],[1 1 1]] [[0],[1],[1]]=`= 0, find x.
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:
[2x 3] `[[1 2],[-3 0]] , [[x],[8]]=0`
`A=[[2,0,1],[2,1,3],[1,-1,0]]` , find A2 − 5A + 4I and hence find a matrix X such that A2 − 5A + 4I + X = 0.
Give examples of matrices
A and B such that AB ≠ BA
Give examples of matrices
A and B such that AB = O but A ≠ 0, B ≠ 0.
Give examples of matrices
A, B and C such that AB = AC but B ≠ C, A ≠ 0.
Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2 B2? Give reasons.
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
For a 2 × 2 matrix A = [aij] whose elements are given by
If A is a square matrix such that A2 = A, then write the value of 7A − (I + A)3, where I is the identity matrix.
If AB = A and BA = B, where A and B are square matrices, then
If A and B are two matrices such n that AB = B and BA = A , `A^2 + B^2` is equal to
If S = [Sij] is a scalar matrix such that sij = k and A is a square matrix of the same order, then AS = SA = ?
If \[A = \begin{bmatrix}\alpha & \beta \\ \gamma & - \alpha\end{bmatrix}\] is such that A2 = I, then
If A = [aij] is a scalar matrix of order n × n such that aii = k, for all i, then trace of A is equal to
(a) nk (b) n + k (c) \[\frac{n}{k}\] (d) none of these
If \[A = \begin{pmatrix}\cos\alpha & - \sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix},\] ,find adj·A and verify that A(adj·A) = (adj·A)A = |A| I3.
If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find X + Y
Give an example of matrices A, B and C such that AB = AC, where A is nonzero matrix, but B ≠ C.
Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2.
If AB = BA for any two square matrices, prove by mathematical induction that (AB)n = AnBn
A matrix which is not a square matrix is called a ______ matrix.
If A and B are square matrices of the same order, then (AB)′ = ______.
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:
- If the number of handmade fans and plates are interchanged for all the schools, then what is the total money collected by all schools?
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?
