Advertisements
Advertisements
Question
Show that AB ≠ BA in each of the following cases
`A=[[-1 1 0],[0 -1 1],[2 3 4]]` and =B `[[1 2 3], [0 1 0],[1 1 0]]`
Solution
`A=[[-1 1 0],[0 -1 1],[2 3 4]]` `[[1 2 3], [0 1 0],[1 1 0]]`
`⇒AB=[[-1+0+0 -2+1+0 -3+0+0],[0+0+1 0-1+1 0+0+0],[2+0+4 4+3+4 6+0+0]]`
`⇒AB=[[-1 -1 -3],[1 0 0],[6 11 6]]`
Also,
`BA= [[1 2 3],[0 1 0],[1 1 0]]``[[-1 1 0],[0 -1 1],[2 3 4]]`...................(1)
`⇒BA = [[-1+0+6 1-2+9 0+2+12],[0+0+0 0-1+0 0+1+0],[-1+0+0 1-1+0 0+1+0]]`
`⇒BA=[[5 8 14],[0 -1 1],[-1 0 1]]`.......................(2)
∴ AB ≠ BA 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
`[(2,3,4),(3,4,5),(4,5,6)][(1,-3,5),(0,2,4), (3,0,5)]`
Compute the indicated product.
`[(3,-1,3),(-1,0,2)][(2,-3),(1,0),(3,1)]`
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 = `[[ cos 2θ sin 2θ],[ -sin 2θ cos 2θ]]`, find A2.
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.
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.
Let A =`[[-1 1 -1],[3 -3 3],[5 5 5]]`and B =`[[0 4 3],[1 -3 -3],[-1 4 4]]`
, compute A2 − B2.
If `A= [[1,2,0],[3,-4,5],[0,-1,3]]` compute A2 − 4A + 3I3.
Find the matrix A such that `[[2,-1],[1,0],[-3,-4]]A` `=[[-1,-8,-10],[1,-2,-5],[9,22,15]]`
Let A and B be square matrices of the same order. Does (A + B)2 = A2 + 2AB + B2 hold? If not, why?
If A and B are square matrices of the same order, explain, why in general
(A + B) (A − B) ≠ A2 − B2
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
(i) Rs 1800
A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received ₹ 2800 as interest. However, if trust had interchanged money in bonds, they would have got ₹ 100 less as interes. Using matrix method, find the amount invested by the trust.
If `A= [[3],[5],[2]]` And B=[1 0 4] , Verify that `(AB)^T=B^TA^T`
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
If \[A = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\] , find A4.
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 \[\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\begin{bmatrix}3 & 1 \\ 2 & 5\end{bmatrix} = \begin{bmatrix}7 & 11 \\ k & 23\end{bmatrix}\] ,then write the value of k.
For a 2 × 2 matrix A = [aij] whose elements are given by
If A and B are two matrices such that AB = A and BA = B, then B2 is equal to
If A and B are two matrices such n that AB = B and BA = A , `A^2 + B^2` 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 \[A = \begin{bmatrix}1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} and B = \begin{bmatrix}1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\] and AB = I3, then x + y equals
The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is
If A and B are square matrices of the same order, then (A + B)(A − B) is equal to
If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find A matrix Z such that X + Y + Z is a zero matrix
If A and B are square matrices of the same order, then (AB)′ = ______.
If A and B are square matrices of the same order, then (kA)′ = ______. (k is any scalar)
A square matrix where every element is unity is called an identity matrix.
If A `= [(1,3),(3,4)]` and A2 − kA − 5I = 0, then the value of k is ______.
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 collected by all three schools DPS, 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 ______.
