Advertisements
Advertisements
Question
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.
Solution
\[Here, \]
\[AB = \begin{bmatrix}0 & c & - b \\ - c & 0 & a \\ b & - a & 0\end{bmatrix}\begin{bmatrix}a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}0 + abc - abc & 0 + b^2 c - b^2 c & 0 + b c^2 - b c^2 \\ - a^2 c + 0 + a^2 c & - abc + 0 + abc & - a c^2 + 0 + a c^2 \\ a^2 b - a^2 b + 0 & a b^2 - a b^2 + 0 & abc - abc + 0\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]
\[ \Rightarrow AB = O_{3 \times 3} . . . \left( 1 \right)\]
\[\]
\[BA = \begin{bmatrix}a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2\end{bmatrix}\begin{bmatrix}0 & c & - b \\ - c & 0 & a \\ b & - a & 0\end{bmatrix}\]
\[ \Rightarrow BA = \begin{bmatrix}0 - abc + abc & a^2 c + 0 - a^2 c & - a^2 b + a^2 b + 0 \\ 0 - b^2 c + b^2 c & abc + 0 - abc & - a b^2 + a b^2 + 0 \\ 0 - b c^2 + b c^2 & a c^2 + 0 - a c^2 & - abc + abc + 0\end{bmatrix}\]
\[ \Rightarrow BA = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]
\[ \Rightarrow BA = O_{3 \times 3} . . . \left( 2 \right)\]
\[ \]
`\ \Rightarrow AB=BA = O_{3 \times 3} ` [From eqs. (1) and (2)]
APPEARS IN
RELATED QUESTIONS
Which of the given values of x and y make the following pair of matrices equal?
`[(3x+7, 5),(y+1, 2-3x)] = [(0,y-2),(8,4)]`
Compute the indicated products:
`[[1 -2],[2 3]][[1 2 3],[-3 2 -1]]`
If A = `[[ab,b^2],[-a^2,-ab]]` , show that A2 = O
If [1 1 x] `[[1 0 2],[0 2 1],[2 1 0]] [[1],[1],[1]]` = 0, find x.
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.
`A=[[3,-5],[-4,2]]` then find A2 − 5A − 14I. Hence, obtain A3
Give examples of matrices
A and B such that AB ≠ BA
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
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.
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?
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 \[A = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}\] , find AAT
If \[A = \begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\] satisfies A4 = λA, then write the value of λ.
If A = [aij] is a 2 × 2 matrix such that aij = i + 2j, write A.
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.
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?
What is the total number of 2 × 2 matrices with each entry 0 or 1?
Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.
If \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & - 1\end{bmatrix}\] , then A2 is equal to ___________ .
If A and B are two matrices such that AB = A and BA = B, then B2 is equal to
If \[\begin{bmatrix}\cos\frac{2\pi}{7} & - \sin\frac{2\pi}{7} \\ \sin\frac{2\pi}{7} & \cos\frac{2\pi}{7}\end{bmatrix}^k = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] then the least positive integral value of k is _____________.
If A, B are square matrices of order 3, A is non-singular and AB = O, then B is a
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 and B are square matrices of the same order, then (A + B)(A − B) is equal to
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 2X – 3Y
If A = `[(3, 5)]`, B = `[(7, 3)]`, then find a non-zero matrix C such that AC = BC.
Give an example of matrices A, B and C such that AB = AC, where A is nonzero matrix, but B ≠ C.
If A = `[(3, -5),(-4, 2)]`, then find A2 – 5A – 14I. Hence, obtain A3.
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?
