Let A 1 2 4 2 1 3 3 4 2 B 3 4 2 2 1 3 1 2 4 C 1 0 6 2 1 3

Let A = [1 2 4 2 1 3 3 4 2] B = [3 4 2 2 1 3 1 2 4] C = [1 0 -6 2 1 3 1 2 4] Find an elementary matrix E that satisfies the given equation a. EA = B b. EB = C c. EB = A d. EC = B

Solution

(a).It may be observed from a scrutiny of A and B that B is derived from A by interchanging the 1^st and the 3^rd rows. The elementary matrix obtained from I₃ by the same operation is E =

0

0

1

0

1

0

1

0

0

Then EA = B.

(b). Let D be the matrix with v₁= (3,4,2)^T,v₂ = (2,1,3)^T, v₃ = (1,2,4)^T and u= (1,0,-6)^T as columns. The RREF of D is

1

0

0

1

0

1

0

0

0

0

1

-2

Thus, the vector u can be obtained by the row operation R₁-2R₃ on the matrix B. The elementary matrix obtained from I₃ by the same operation is E =

1

0

-2

0

1

0

0

0

1

Then EB =C.

(c ). It may be observed from a scrutiny of A and B that A is derived from B by interchanging the 1^st and the 3^rd rows. The elementary matrix obtained from I₃ by the same operation is E =

0

0

1

0

1

0

1

0

0

Then EB = A

(d). From part (b) above, we know that B is derived from C by the row operation R₁ +2R₃ on the matrix C. The elementary matrix obtained from I₃ by the same operation is E =

1

0

2

0

1

0

0

0

1

Then EC = B.

0	0	1
0	1	0
1	0	0