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

Solutions

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 ?


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