Advertisements
Advertisements
प्रश्न
If A = `[[0,c,-b],[-c,0,a],[b,-a,0]]`and B =`[[a^2 ,ab,ac],[ab,b^2,bc],[ac,bc,c^2]]`, show that AB = BA = O3×3.
उत्तर
\[Here, \]
\[AB = \begin{bmatrix}0 & c & - b \\ - c & 0 & a \\ b & - a & 0\end{bmatrix}\begin{bmatrix}a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}0 + abc - abc & 0 + b^2 c - b^2 c & 0 + b c^2 - b c^2 \\ - a^2 c + 0 + a^2 c & - abc + 0 + abc & - a c^2 + 0 + a c^2 \\ a^2 b - a^2 b + 0 & a b^2 - a b^2 + 0 & abc - abc + 0\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]
\[ \Rightarrow AB = O_{3 \times 3} . . . \left( 1 \right)\]
\[\]
\[BA = \begin{bmatrix}a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2\end{bmatrix}\begin{bmatrix}0 & c & - b \\ - c & 0 & a \\ b & - a & 0\end{bmatrix}\]
\[ \Rightarrow BA = \begin{bmatrix}0 - abc + abc & a^2 c + 0 - a^2 c & - a^2 b + a^2 b + 0 \\ 0 - b^2 c + b^2 c & abc + 0 - abc & - a b^2 + a b^2 + 0 \\ 0 - b c^2 + b c^2 & a c^2 + 0 - a c^2 & - abc + abc + 0\end{bmatrix}\]
\[ \Rightarrow BA = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]
\[ \Rightarrow BA = O_{3 \times 3} . . . \left( 2 \right)\]
\[ \]
`\ \Rightarrow AB=BA = O_{3 \times 3} ` [From eqs. (1) and (2)]
APPEARS IN
संबंधित प्रश्न
Compute the indicated product:
`[(a,b),(-b,a)][(a,-b),(b,a)]`
Compute the indicated product.
`[(1, -2),(2,3)][(1,2,3),(2,3,1)]`
Compute the products AB and BA whichever exists in each of the following cases:
`A=[[3 2],[-1 0],[-1 1]]` and `B= [[4 5 6],[0 1 2]]`
Compute the products AB and BA whichever exists in each of the following cases:
A = [1 −1 2 3] and B=`[[0],[1],[3],[2]]`
If A = `[[4 2],[-1 1]]`
, prove that (A − 2I) (A − 3I) = O
If A = `[[1 1],[0 1]]` show that A2 = `[[1 2],[0 1]]` and A3 = `[[1 3],[0 1]]`
For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:
`A=[[2 -1],[1 1],[-1 2]]` `B=[[0 1],[1 1]]` C=`[[1 -1],[0 1]]`
\[A = \begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}\] , Show that A2 = A.
If [1 −1 x] `[[0 1 -1],[2 1 3],[1 1 1]] [[0],[1],[1]]=`= 0, find x.
If \[A = \begin{bmatrix}3 & - 5 \\ - 4 & 2\end{bmatrix}\] , find A2 − 5A − 14I.
Solve the matrix equations:
`[1 2 1] [[1,2,0],[2,0,1],[1,0 ,2]][[0],[2],[x]]=0`
`A=[[1,2,2],[2,1,2],[2,2,1]]`, then prove that A2 − 4A − 5I = 0
Find a 2 × 2 matrix A such that `A=[[1,-2],[1,4]]=6l_2`
`A=[[3,-5],[-4,2]]` then find A2 − 5A − 14I. Hence, obtain A3
If `P=[[x,0,0],[0,y,0],[0,0,z]]` and `Q=[[a,0,0],[0,b,0],[0,0,c]]` prove that `PQ=[[xa,0,0],[0,yb,0],[0,0,zc]]=QP`
\[A = \begin{bmatrix}\cos \alpha + \sin \alpha & \sqrt{2}\sin \alpha \\ - \sqrt{2}\sin \alpha & \cos \alpha - \sin \alpha\end{bmatrix}\] ,prove that
\[A^n = \begin{bmatrix}\text{cos n α} + \text{sin n α} & \sqrt{2}\text{sin n α} \\ - \sqrt{2}\text{sin n α} & \text{cos n α} - \text{sin n α} \end{bmatrix}\] for all n ∈ 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)2 ≠ A2 − 2AB + B2
Let `A=[[1,1,1],[3,3,3]],B=[[3,1],[5,2],[-2,4]]` and `C=[[4,2],[-3,5],[5,0]]`Verify that AB = AC though B ≠ C, A ≠ O.
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
\[A = \begin{bmatrix}140 \\ 200 \\ 150\end{bmatrix}\begin{array} \text{Telephone}\\{\text{House calls }}\\ \text{Letters}\end{array}\]
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?
If `A= [[3],[5],[2]]` And B=[1 0 4] , Verify that `(AB)^T=B^TA^T`
If A is an m × n matrix and B is n × p matrix does AB exist? If yes, write its order.
Given an example of two non-zero 2 × 2 matrices A and B such that AB = O.
If \[A = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}\] , find AAT
Write matrix A satisfying ` A+[[2 3],[-1 4]] =[[3 6],[- 3 8]]`.
If A is 2 × 3 matrix and B is a matrix such that AT B and BAT both are defined, then what is the order of B ?
If A, B are square matrices of order 3, A is non-singular and AB = O, then B is a
If \[A = \begin{bmatrix}\alpha & \beta \\ \gamma & - \alpha\end{bmatrix}\] is such that A2 = I, then
If A and B are square matrices of the same order, then (A + B)(A − B) is equal to
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.
If AB = BA for any two square matrices, prove by mathematical induction that (AB)n = AnBn
If A = `[(0, 1),(1, 0)]`, then A2 is equal to ______.
If matrix A = [aij]2×2, where aij `{:(= 1 "if i" ≠ "j"),(= 0 "if i" = "j"):}` then A2 is equal to ______.
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:
- How many articles (in total) are sold by three schools?
Let a, b, c ∈ R be all non-zero and satisfy a3 + b3 + c3 = 2. If the matrix A = `((a, b, c),(b, c, a),(c, a, b))` satisfies ATA = I, then a value of abc can be ______.
