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Question 45392This question is from textbook College Algebra
: Find the inverse of each matrix A if possible. Check that AA^-1=I and A^-1 A=I.
[1 -1 2]
[1 2 3]
[2 1 5]
This question is from textbook College Algebra
Answer by AnlytcPhil(1807)   (Show Source):
You can put this solution on YOUR website! Find the inverse of each matrix A if possible.
Check that AA^-1=I and A^-1 A=I.
[1 -1 2]
[1 2 3]
[2 1 5]
Augment the matrix with the identity matrix:
[ 1 -1 2 | 1 0 0]
[ 1 2 3 | 0 1 0]
[ 2 1 5 | 0 0 1]
The idea is to get every element in the
left side 0 except the three diagonal elements,
which must not be 0. Then we divide each row
through to make the diagonal elements on the
left side 1.
Get a 0 where the 1 is in the 2nd row 1st column by
multiplying row 1 by -1 and adding it to row 2,
restoring row 1
-1[ 1 -1 2 | 1 0 0]
1[ 1 2 3 | 0 1 0]
[ 2 1 5 | 0 0 1]
[ 1 -1 2 | 1 0 0]
[ 0 3 1 | -1 1 0]
[ 2 1 5 | 0 0 1]
Get a 0 where the 2 is in the 3rd row 1st column by
multiplying row 1 by -2 and adding it to row 3,
restoring row 1
-2[ 1 -1 2 | 1 0 0]
[ 0 3 1 | -1 1 0]
1[ 2 1 5 | 0 0 1]
[ 1 -1 2 | 1 0 0]
[ 0 3 1 | -1 1 0]
[ 0 3 1 | -2 0 1]
Get a 0 where the -1 is in the 1st row 2nd column by
multiplying row 2 by 1 and adding it to 3 times row 1,
restoring row 2
3[ 1 -1 2 | 1 0 0]
1[ 0 3 1 | -1 1 0]
[ 0 3 1 | -2 0 1]
[ 3 0 7 | 2 1 0]
[ 0 3 1 | -1 1 0]
[ 0 3 1 | -2 0 1]
Get a 0 where the 3 is in the 3rd row 2nd column by
multiplying row 2 by -1 and adding it to 1 times row 3,
restoring row 2
[ 3 0 7 | 2 1 0]
-1[ 0 3 1 | -1 1 0]
1[ 0 3 1 | -2 0 1]
[ 3 0 7 | 2 1 0]
[ 0 3 1 | -1 1 0]
[ 0 0 0 | -2 0 1]
Oh, oh. The element in the 3rd row 2rd column is a 0
so there is no way to get 0's where the 7 and 1 are
above it. This matrix has no inverse.
Edwin
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