Determine all 2×2 matrices A such that AB = BA for any 2×2 matrix B.

Question

hesgidiauE · Accepted Answer

Step 1  Let A=[abcd] and B=[pqrs] be any 2×2  matrices.  Step 5  Find AB.  AB=[abcd][pqrs]  =[ap+braq+bscp+drcq+ds]  Find BA.  BA=[pqrs][abcd]  =[ap+cqbp+dqar+csbr+ds] Equate the matrices AB=BA.  AB=BA  [ap+braq+bscp+drcq+ds]=[ap+cqbp+dqar+csbr+ds]  [ap+braq+bscp+drcq+ds]−[ap+cqbp+dqar+csbr+ds]=[0000]  [br−cq(a−d)q−b(p−s)c(p−s)−r(a−d)cq−br]=[0000]  Equate the matrices.  br−cq=0…(1)  (a−d)q−b(p−s)=0…(2)  c(p−s)−r(a−d)=0…(3)  cq−br=0…(4)  From equation (1) and equation (4),  q=bcr  Substitute q=bcr in equation (2)  (a−d)bce−b(p−s)=0  c=(a−d)r(p−s)  Substitute q=bcr in equation (2)  (a−d)(bcr)−b(p−s)=0  b=q(a−d)p−s  Hence, the 2x2 matrix which satisfied AB=BA for any matrix B is   A=[aq(a−d)p−sr(a−d)p−sd]

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Nick Camelot · Accepted Answer

Answer:A=[abcd]=[xyzw]Explanation:Given:A=[abcd] where a, b, c, and d are the entries of the matrix A.Now, let&#039;s consider an arbitrary 2×2 matrix B:B=[xyzw]To satisfy the condition AB = BA for any matrix B, we multiply the matrices AB and BA and set them equal:AB=[abcd][xyzw]=[ax+bzay+bwcx+dzcy+dw]BA=[xyzw][abcd]=[xa+ycxb+ydza+wczb+wd]Setting AB = BA, we have:[ax+bzay+bwcx+dzcy+dw]=[xa+ycxb+ydza+wczb+wd]By comparing the corresponding entries, we can obtain the following equations:1) ax+bz=xa+yc2) ay+bw=xb+yd3) cx+dz=za+wc4) cy+dw=zb+wdSimplifying these equations, we get:1) ax−xa=yc−bz⟹ax−xa−yc+bz=0⟹(a−x)(x−a)+(b−y)(y−b)=0⟹(a−x)2+(b−y)2=02) ay−yd=xb−bw⟹(a−x)(y−d)+(b−y)(w−b)=0⟹(a−x)(y−d)+(b−y)(b−w)=03) cx−za=wc−dz⟹(c−z)(x−a)+(d−z)(w−c)=0⟹(c−z)(x−a)+(d−z)(d−w)=04) cy−zb=wd−dw⟹(c−z)(y−d)+(d−z)(d−w)=0⟹(c−z)(y−d)+(d−z)(w−d)=0Now, let&#039;s analyze each equation:Equation 1:(a−x)2+(b−y)2=0For this equation to hold true for all values of x and y, we must have a = x and b = y. Therefore, the entries of A are fixed:a=xb=yEquation 2:(a−x)(y−d)+(b−y)(b−w)=0Since a = x and b = y, this equation simplifies to:(x−x)(y−d)+(y−y)(y−w)=00(y−d)+0(y−w)=0This equation holds true for any values of d and w.Equation 3:(c−z)(x−a)+(d−z)(d−w)=0Substituting a = x and c = z, we get:(z−z)(x−x)+(d−z)(d−w)=00(x−x)+(d−z)(d−w)=0This equation holds true for any values of x, z, d, and w.Equation 4:(c−z)(y−d)+(d−z)(w−d)=0Substituting c = z and d = w, we obtain:(c−c)(y−d)+(w−w)(w−d)=00(y−d)+0(w−d)=0This equation holds true for any values of y, d, and w.Therefore, the general form of the matrix A that satisfies the condition AB = BA for any matrix B is:A=[abcd]=[xyzw] where a = x, b = y, c = z, and d = w.

Determine all 2 times 2 matrices A such that AB = BA for any 2 times 2 matrix B.

Answered question

Answer & Explanation

New Questions in Precalculus