Question

In: Advanced Math

solve for matrix B Let I be Identity matrix (I-2B)-1= 1 -3 3 -2 2 -5...

solve for matrix B

Let I be Identity matrix

(I-2B)^-1=

1	-3	3
-2	2	-5
3	-8	9

Expert Solution

Matrix I is

A1 A2 A3

1 1 0 0

2 0 1 0

3 0 0 1

Matrix 2B is

B1 B2 B3

1 1 -6 6

2 -4 4 -10

3 6 -16 18

now on subtracting 2B from I ,we get following matrix

C1 C2 C3

1 0 6 -6

2 4 -3 10

3 -6 16 -17

now we have find inverse of above matrix

Your matrix is

	A1	A2	A3
1	0	6	-6
2	4	-3	10
3	-6	16	-17

Determinant is not zero, therefore inverse matrix exists

Write the augmented matrix

	A1	A2	A3	B1	B2	B3
1	0	6	-6	1	0	0
2	4	-3	10	0	1	0
3	-6	16	-17	0	0	1

Make the pivot in the 1st column by dividing the 2nd row by 4 and swap the 2nd and the 2nd rows

	A1	A2	A3	B1	B2	B3
1	1	-0.75	2.5	0	0.25	0
2	0	6	-6	1	0	0
3	-6	16	-17	0	0	1

Multiply the 1st row by -6

	A1	A2	A3	B1	B2	B3
1	-6	4.5	-15	0	-1.5	0
2	0	6	-6	1	0	0
3	-6	16	-17	0	0	1

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

	A1	A2	A3	B1	B2	B3
1	1	-0.75	2.5	0	0.25	0
2	0	6	-6	1	0	0
3	0	11.5	-2	0	1.5	1

Make the pivot in the 2nd column by dividing the 2nd row by 6

	A1	A2	A3	B1	B2	B3
1	1	-0.75	2.5	0	0.25	0
2	0	1	-1	0.16666666666666666666	0	0
3	0	11.5	-2	0	1.5	1

Multiply the 2nd row by -0.75

	A1	A2	A3	B1	B2	B3
1	1	-0.75	2.5	0	0.25	0
2	0	-0.75	0.75	-0.125	0	0
3	0	11.5	-2	0	1.5	1

Subtract the 2nd row from the 1st row and restore it

	A1	A2	A3	B1	B2	B3
1	1	0	1.75	0.125	0.25	0
2	0	1	-1	0.16666666666666666666	0	0
3	0	11.5	-2	0	1.5	1

Multiply the 2nd row by 11.5

	A1	A2	A3	B1	B2	B3
1	1	0	1.75	0.125	0.25	0
2	0	11.5	-11.5	1.9166666666666666665	0	0
3	0	11.5	-2	0	1.5	1

Subtract the 2nd row from the 3rd row and restore it

	A1	A2	A3	B1	B2	B3
1	1	0	1.75	0.125	0.25	0
2	0	1	-1	0.16666666666666666666	0	0
3	0	0	9.5	-1.9166666666666666665	1.5	1

Make the pivot in the 3rd column by dividing the 3rd row by 9.5

	A1	A2	A3	B1	B2	B3
1	1	0	1.75	0.125	0.25	0
2	0	1	-1	0.16666666666666666666	0	0
3	0	0	1	-0.20175438596491228068	0.15789473684210526315	0.1052631578947368421

Multiply the 3rd row by 1.75

	A1	A2	A3	B1	B2	B3
1	1	0	1.75	0.125	0.25	0
2	0	1	-1	0.16666666666666666666	0	0
3	0	0	1.75	-0.35307017543859649119	0.27631578947368421051	0.18421052631578947367

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

	A1	A2	A3	B1	B2	B3
1	1	0	0	0.47807017543859649119	-0.02631578947368421051	-0.18421052631578947367
2	0	1	-1	0.16666666666666666666	0	0
3	0	0	1	-0.20175438596491228068	0.15789473684210526315	0.1052631578947368421

Multiply the 3rd row by -1

	A1	A2	A3	B1	B2	B3
1	1	0	0	0.47807017543859649119	-0.02631578947368421051	-0.18421052631578947367
2	0	1	-1	0.16666666666666666666	0	0
3	0	0	-1	0.20175438596491228068	-0.15789473684210526315	-0.1052631578947368421

Subtract the 3rd row from the 2nd row and restore it

	A1	A2	A3	B1	B2	B3
1	1	0	0	0.47807017543859649119	-0.02631578947368421051	-0.18421052631578947367
2	0	1	0	-0.03508771929824561402	0.15789473684210526315	0.1052631578947368421
3	0	0	1	-0.20175438596491228068	0.15789473684210526315	0.1052631578947368421

There is the inverse matrix on the right

	A1	A2	A3	B1	B2	B3
1	1	0	0	0.47807017543859649119	-0.02631578947368421051	-0.18421052631578947367
2	0	1	0	-0.03508771929824561402	0.15789473684210526315	0.1052631578947368421
3	0	0	1	-0.20175438596491228068	0.15789473684210526315	0.1052631578947368421

Result is

	B1	B2	B3
1	0.47807017543859649119	-0.02631578947368421051	-0.18421052631578947367
2	-0.03508771929824561402	0.15789473684210526315	0.1052631578947368421
3	-0.20175438596491228068	0.15789473684210526315	0.1052631578947368421

please like ?

Colby Messinger answered 1 year ago

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