Quiz7S11mt210Sample2Ans

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Published on 31 Jan 2019
Department
Course
Professor
MT Q  S 
İ S. Y
M , 
Surname, Name:
Q . §. CHARACTERIZATION OF INVERSE MATRICES
Let T:R3ÏR3be the linear transformation defined by
(x1, x2, x3)(x1+x2+x3, x1x2x3, x1x2+x3).
Is Tan invertible transformation? If it is, find T1.
A
REMARK. A linear transformation Tis invertible if and only if the standard
matrix Aof Tis invertible. The standard matrix of Tis A=
1 1 1
111
11 1
.
We see that
1 1 1
111
11 1
100
010
001
(1)R1+R2R2
(1)R1+R3R3
//
1 1 1
022
02 0
1 0 0
1 1 0
1 0 1
(1)R2+R3R3
//
1 1 1
022
0 0 2
1 0 0
1 1 0
01 1
(1)R3+R2R3
(1
2)R3+R1R1
//
1 1 0
02 0
0 0 2
11
21
2
1 0 1
01 1
(1
2)R3R3
(1
2)R2R2
//
1 1 0
0 1 0
0 0 1
11
21
2
1
201
2
01
2
1
2
(1)R2+R1R1
//
100
010
001
1
2
1
20
1
201
2
01
2
1
2
.
We observe that Ais invertible and A1=
1
2
1
20
1
201
2
01
2
1
2
.Thus, Tis in-
vertible and T1(x) = A1xor
T1(x1, x2, x3) = (1
2x1+1
2x2,1
2x1
1
2x3,
1
2x2+1
2x3).
1
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