SOLUTION: find the inverse matrix of the following. Show your work A= 1 1 2 1 1 2 2 1 -2 -2 -3 0 0 1 2 3

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Question 1102859: find the inverse matrix of the following. Show your work


A= 1 1 2 1
1 2 2 1
-2 -2 -3 0
0 1 2 3
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
Start with the matrix augmented by the 4x4 identity matrix
on the RIGHT:



Then do row operations until you have the 4x4 identity matrix
on the LEFT, like this:

  

Then the answer is the 4x4 matrix on the right:

  

If you have trouble filling in the steps, tell me in the
message form below and I'll get back to you by email to
help you with them.

Edwin
Answer by ikleyn(52810)   (Show Source): You can put this solution on YOUR website!
.
Your matrix

        A1	A2	A3	A4
1	1	1	2	1
2	1	2	2	1
3	-2	-2	-3	0
4	0	1	2	3

Determinant is not zero, therefore inverse matrix exists


Write the augmented matrix

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	1	2	1	1	0	0	0
2	1	2	2	1	0	1	0	0
3	-2	-2	-3	0	0	0	1	0
4	0	1	2	3	0	0	0	1

Find the pivot in the 1st column in the 1st row

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	1	2	1	1	0	0	0
2	1	2	2	1	0	1	0	0
3	-2	-2	-3	0	0	0	1	0
4	0	1	2	3	0	0	0	1

Subtract the 1st row from the 2nd row

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	1	2	1	1	0	0	0
2	0	1	0	0	-1	1	0	0
3	-2	-2	-3	0	0	0	1	0
4	0	1	2	3	0	0	0	1

Multiply the 1st row by -2

       A1	A2	A3	A4	B1	B2	B3	B4
1	-2	-2	-4	-2	-2	0	0	0
2	0	1	0	0	-1	1	0	0
3	-2	-2	-3	0	0	0	1	0
4	0	1	2	3	0	0	0	1

Subtract the 1st row from the 3rd row and restore it

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	1	2	1	1	0	0	0
2	0	1	0	0	-1	1	0	0
3	0	0	1	2	2	0	1	0
4	0	1	2	3	0	0	0	1

Find the pivot in the 2nd column in the 2nd row

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	1	2	1	1	0	0	0
2	0	1	0	0	-1	1	0	0
3	0	0	1	2	2	0	1	0
4	0	1	2	3	0	0	0	1

Subtract the 2nd row from the 1st row

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	2	1	2	-1	0	0
2	0	1	0	0	-1	1	0	0
3	0	0	1	2	2	0	1	0
4	0	1	2	3	0	0	0	1

Subtract the 2nd row from the 4th row

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	2	1	2	-1	0	0
2	0	1	0	0	-1	1	0	0
3	0	0	1	2	2	0	1	0
4	0	0	2	3	1	-1	0	1

Find the pivot in the 3rd column in the 3rd row

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	2	1	2	-1	0	0
2	0	1	0	0	-1	1	0	0
3	0	0	1	2	2	0	1	0
4	0	0	2	3	1	-1	0	1

Multiply the 3rd row by 2

A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	2	1	2	-1	0	0
2	0	1	0	0	-1	1	0	0
3	0	0	2	4	4	0	2	0
4	0	0	2	3	1	-1	0	1

Subtract the 3rd row from the 1st row

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	0	-3	-2	-1	-2	0
2	0	1	0	0	-1	1	0	0
3	0	0	2	4	4	0	2	0
4	0	0	2	3	1	-1	0	1

Subtract the 3rd row from the 4th row and restore it

A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	0	-3	-2	-1	-2	0
2	0	1	0	0	-1	1	0	0
3	0	0	1	2	2	0	1	0
4	0	0	0	-1	-3	-1	-2	1

Find the pivot in the 4th column in the 4th row (inversing the sign in the whole row)

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	0	-3	-2	-1	-2	0
2	0	1	0	0	-1	1	0	0
3	0	0	1	2	2	0	1	0
4	0	0	0	1	3	1	2	-1

Multiply the 4th row by -3

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	0	-3	-2	-1	-2	0
2	0	1	0	0	-1	1	0	0
3	0	0	1	2	2	0	1	0
4	0	0	0	-3	-9	-3	-6	3

Subtract the 4th row from the 1st row and restore it

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	0	0	7	2	4	-3
2	0	1	0	0	-1	1	0	0
3	0	0	1	2	2	0	1	0
4	0	0	0	1	3	1	2	-1

Multiply the 4th row by 2

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	0	0	7	2	4	-3
2	0	1	0	0	-1	1	0	0
3	0	0	1	2	2	0	1	0
4	0	0	0	2	6	2	4	-2

Subtract the 4th row from the 3rd row and restore it

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	0	0	7	2	4	-3
2	0	1	0	0	-1	1	0	0
3	0	0	1	0	-4	-2	-3	2
4	0	0	0	1	3	1	2	-1

There is the inverse matrix on the right

       A1	A2	A3	A4	B1	B2	B3	B4
1	1	0	0	0	7	2	4	-3
2	0	1	0	0	-1	1	0	0
3	0	0	1	0	-4	-2	-3	2
4	0	0	0	1	3	1	2	-1

Result:

       B1	B2	B3	B4
1	7	2	4	-3
2	-1	1	0	0
3	-4	-2	-3	2
4	3	1	2	-1


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