Let A 3 1 2 2 3 1 1 2 3 and b b1 b2 b3 Let Bj be the matric

Let A= [3 1 2 2 3 1 1 2 3] and b = [b_1 b_2 b_3]. Let B_j be the matrices formed according to Cramer\'s rule with the j-th column of A replaced by b. If det B_1 = 18, det B_2 = -18, and det B_3 = 36 then what is b? Find b_1, b_2 and b_3.

Solution

Determinant of A = 3(7) -1(5)+2(1) = 21-5 +2= 18

the matrix eqn is AX=b where X =(x₁ ,x₂,x₃)^T adn by Crammers Rule

x₁ = det B₁/detA = 18/18=1

x₂=detB₂/detA =-18/18=-1 and x₃= detB₃/detA =36/18 =2

substituting the values of x₁,x₂,x₃ we get the matrix b

AX=b => 3x₁+x₂+2x₃=b_{1 ie 3-1+4=} 6=b1

  2x₁+3x₂+x₃=b_{2 i}ie 2-3+2=1=b₂

x₁+2x₂+3x₃=b₃   ie 1-2+6=5=b₃