Home
Class 12
MATHS
If the matrices A=[(1,2),(3,4)] and B=[(...

If the matrices `A=[(1,2),(3,4)] and B=[(a,b),(c,d)] (a,b,cd` not all simultaneously zero) commute, find the value of `(d-b)/(a+c-b).` Also show that the matrix which commutes with A is of the form `[(alpha-beta,(2beta)/3),(beta,alpha)]`

Text Solution

Verified by Experts

Given, AB = BA
`[[1,2],[3,4]] [[a,b],[c,d]]=[[a,b],[c,d]][[1,2],[3,4]]`
`rArr[[a+2c,b+2d],[3a+4c,3b+4d]]=[[a+3b,2a+4b],[c+3d,2c+4d]] `
On comparing, we get
`a + 2c = a +3b`
`rArr b= (2c)/3` …(i)
`b + 2d =2a +2b`
`rArr d = a+(3b)/2 ` ... (ii)
`rArr 3a + 4c= c + 3d`
`d = a +c` ...(iii)
and `3b + 4 d = 3c + 4d`
`rArr b = 2c/3` ...(iv)
`rArr (d-b)/(a+c-b)=(d-b)/(d-b) = 1 ` [frp, Eq. (iii)] Now, `B= [[a,b],[c,d]]= [[d-c,(2c)/3],[c,d]]`
If `c= beta and d= alpha,` then `B= [[alpha-beta ,(2beta)/3],[beta,alpha]]`
Promotional Banner

Similar Questions

Explore conceptually related problems

Let A=[(0,alpha),(0,0)] and (A+I)^(50)-50 A=[(a,b),(c,d)] Then the value of a+b+c+d is

If alpha and beta are zeros of f(x) = 6x^(2) + x= 2 find the value fo (alpha)/(beta) +(beta)/(alpha) .

Let A=[(2,-1),(3,4)], B=[(5,2),(7,4)], C=[(2,5),(3,8)] . Find a matrix D such that CD=AB .

If a point (alpha, beta) lies on the circle x^(2)+y^(2)=1 , then the locus of the point, (3 alpha+2, beta) is

If x^3+3x^2-9x+c is of the form (x-alpha)^2(x-beta) then c is equal to

If alpha, beta are the roots of a x^(2)+b x+c=0 the equation whose roots are alpha+(1)/(beta) and beta+(1)/(alpha) is

Let alpha, beta the roots of the equation (x - a) (x - b) = c , c ne 0. Then the roots of the equation (x - alpha ) x - beta) + c = 0 are :

If alpha and beta are the zeroes of the quadratic polynomial 2-3x-x^(2) then what is the value of alpha+beta+alpha beta ?

alpha+beta are the zeroes of the polynomial x^(2)-6x+4 , then the value of ((alpha+beta)^(2)-2alpha beta)/(alpha beta) is