Now let\'s apply this idea to find the atrix of au inverse transfornnation 9. (5
ID: 3135914 • Letter: N
Question
Now let's apply this idea to find the atrix of au inverse transfornnation 9. (5.3.5) Let T be a linear transformation induced by the matrix A = Fiud the uatrix of T- ). Let T be a linear transformation induced by the matrix A 0 3 0 0 0 3 Then T dilates space by a factor of 3. Find the matrix A-1 of T-1. What does the trausformation T-1 do? Remember that inverso matrices multiply to be the identity: AA-1I. So this is how you carn check that you did indeed find the right matrix. If they don't multiply to he the identity then they are not inverses! Show that you did indeed find the right inverse by showing that their product equals the 3x3 identity matrix I.Explanation / Answer
9. (5.3.5). The linear transformation T is represented by the matrix A =
2
1
5
2
The inverse linear transformation T-1 is represented by the matrix A-1.
Let M =
2
1
1
0
5
2
0
1
To find A-1, we will reduce M to its RREF as under:
Multiply the 1st row by ½
Add -5 times the 1st row to the 2nd row
Multiply the 2nd row by -2
Add -1/2 times the 2nd row to the 1st row
Then the RREF of M is
1
0
-2
1
0
1
5
-2
Hence A-1 =
-2
1
5
-2
It may be observed that AA-1= A-1 A = I2
10. Let M =
3
0
0
1
0
0
0
3
0
0
1
0
0
0
3
0
0
1
To find A-1, we will reduce M to its RREF as under:
Multiply the 1st row by 1/3
Multiply the 2nd row by 1/3
Multiply the 3rd row by 1/3
Then the RREF of M is
1
0
0
1/3
0
0
0
1
0
0
1/3
0
0
0
1
0
0
1/3
Hence A-1 =
1/3
0
0
0
1/3
0
0
0
1/3
It may be observed that AA-1= A-1 A = I3.
The inverse linear transformation T-1 is represented by the matrix A-1. It shrinks space to 1/3 rd of its earlier size.
2
1
5
2
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