Published on 31 Jan 2019

School

Department

Course

Professor

MT Q

İ S. Y

M ,

Surname, Name:

Q . §. C I M

Let T:R3ÏR3be the linear transformation deﬁned by

(x1, x2, x3)7Ï (x1+x2+x3, x2+x3, x3).

Let S:R3ÏR3be the linear transformation deﬁned by

(x1, x2, x3)7Ï (x3, x2+x3, x1+x2+x3).

(A) Is S◦Tinvertible? If it is, ﬁnd the formula for (S◦T)−1.

(B) Is T◦Sinvertible? If it is, ﬁnd the formula for (T◦S)−1.

A

(A) The standard matrices Aand Bof Tand S, respectively, are

A=

111

011

001

and B=

0 0 1

0 1 1

1 1 1

.

Note that A∼I3×3and B∼I3×3. Therefore, both Aand Bare invertible which

means that S◦Tand T◦Sare invertible. By using the algorithm for inverse

matrices, we calculate the inverses as

A−1=

1−1 0

0 1 −1

0 0 1

and B−1=

0−1 1

−1 1 0

1 0 0

.

The standard matrix of (S◦T)−1=T−1◦S−1is given by the multiplication

A−1B−1. Thus, the formula for (S◦T)−1is obtained as

(S◦T)−1(x) =

1−1 0

0 1 −1

0 0 1

0−1 1

−1 1 0

1 0 0

x1

x2

x3

=

1−2 1

−2 1 0

1 0 0

x1

x2

x3

.

We get (S◦T)−1(x) = (x1−2x2+x3,−2x1+x2, x1).

(B) Similarly, the standard matrix of (T◦S)−1=S−1◦T−1is given by the

multiplication B−1A−1. Thus, the formula for (S◦T)−1is obtained as

(T◦S)−1(x) =

0−1 1

−1 1 0

1 0 0

1−1 0

0 1 −1

0 0 1

x1

x2

x3

=

0−1 2

−1 2 −1

1−1 0

x1

x2

x3

.

We get (T◦S)−1(x) = (−x2+ 2x3,−x1+ 2x2−x3, x1−x2).

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