Advertisements
Advertisements
Question
If A =
\[\begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}\]and B =
\[\begin{bmatrix}- 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5\end{bmatrix}\] , show that AB = BA = O3×3.
Solution
Here ,
AB= `[[2 -3 -5],[-1 4 5],[1 -3 -4]]``[[-1 3 5],[1 -3 -5],[-1 3 5]]`
`⇒AB =[[-2-3+5 6+9- 15 10+15-25],[1+4-5 -3-12+15 -5-20=25],[-1-3+4 3+9-12 5+15-20]]`
`⇒AB=[[0 0 0],[0 0 0],[0 0 0]]`
`⇒AB = O3xx3` .................(1)
`⇒BA=[[-1 3 5],[1 -3 -5],[-1 3 5]]``[[2 -3 -5],[-1 4 5],[1 -3 -4]]`
`⇒BA=[[-2-3+5 3+12-15 5+15-20],[2+3-5 -3-12+15 -5-15+20],[-2-3+5 3+12-15 5+15-20]]`
`⇒BA=[[0 0 0 ],[0 0 0],[0 0 0]]`
`⇒BA=0_3×3` ...(2)
`⇒AB=BA=0_3×3 ` [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 product.
`[(1),(2),(3)] [2,3,4]`
Compute the indicated product.
`[(1, -2),(2,3)][(1,2,3),(2,3,1)]`
Compute the indicated product.
`[(3,-1,3),(-1,0,2)][(2,-3),(1,0),(3,1)]`
A trust fund has Rs. 30,000 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. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of Rs 2,000.
Compute the indicated products:
`[[a b],[-b a]][[a -b],[b a]]`
Compute the indicated products:
`[[1 -2],[2 3]][[1 2 3],[-3 2 -1]]`
Evaluate the following:
`([[1 3],[-1 -4]]+[[3 -2],[-1 1]])[[1 3 5],[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=[[4 2 3],[1 1 2],[3 0 1]]`=`B=[[1 -1 1],[0 1 2],[2 -1 1]]` and `C= [[1 2 -1],[3 0 1],[0 0 1]]`
If A= `[[1 0 -2],[3 -1 0],[-2 1 1]]` B=,`[[0 5 -4],[-2 1 3],[-1 0 2]] and C=[[1 5 2],[-1 1 0],[0 -1 1]]` verify that A (B − C) = AB − AC.
Compute the elements a43 and a22 of the matrix:`A=[[0 1 0],[2 0 2],[0 3 2],[4 0 4]]` `[[2 -1],[-3 2],[4 3]] [[0 1 -1 2 -2],[3 -3 4 -4 0]]`
If A=then find λ, μ so that A2 = λA + μI
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.
If , then show that A is a root of the polynomial f (x) = x3 − 6x2 + 7x + 2.
Find the matrix A such that `[[2,-1],[1,0],[-3,-4]]A` `=[[-1,-8,-10],[1,-2,-5],[9,22,15]]`
Find a 2 × 2 matrix A such that `A=[[1,-2],[1,4]]=6l_2`
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`
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)2 ≠ A2 − 2AB + B2
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?
The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹ 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?
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 (2A)T = 2AT.
write AB.
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.
Write a 2 × 2 matrix which is both symmetric and skew-symmetric.
If A and B are two matrices such n that AB = B and BA = A , `A^2 + B^2` is equal to
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
The matrix \[A = \begin{bmatrix}0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0\end{bmatrix}\] is a
If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find 2X – 3Y
Give an example of matrices A, B and C such that AB = AC, where A is nonzero matrix, but B ≠ C.
If A = `[(1, 1, 1),(0, 1, 1),(0, 0, 1)]` and M = A + A2 + A3 + .... + A20, then the sum of all the elements of the matrix M is equal to ______.
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
