3. Solve the following systems of equations using Gaussian elimination. (a) 2x₁ x1 - - 3x2 + 2x3 = 0 x2 + x3 = 7 (b) -x₁ 2 x2 + x3 = 2x + 2x2 - 4.x3 = -4 -x, +5x2+4x3 = 4 x₁ - 2x2 + 3x3 = (c)-x - 3x2 + 2x3 = -2 (d) 2x + 4x2 - 2x3 = 2x + x2 + 3x3 = % 5x+4x2+6x3 + 12 - 2x2 - 4x3 = -2x1 - x2-3x3 = -4 (e) x + x + 4x3 = 4 (f) 2x-3x2 - x3 = 2 2x,+2+3x3 = 5 3.x₁ - 5.x2 5x₁ + 2x2 + 5x3 = 11 = - 2x3 9x1 + 6x2 + 4x3 = -1 4. Solve the problems in Exercise 3 using elimination by pivoting (Gauss-Jordan elimination). 5. (a) Write the LU decomposition for each coefficient matrix A in Ex- ercise 3. (b) Multiply L times U to show that the product is A, for each coef- ficient matrix A in Exercise 3. 6. Find the determinant of each matrix in Exercise 3 using Theorem 2. 7. Re-solve each system in Exercise 3 with the new right-hand-side vector [10, 5, 10] using the numbers in the L and U matrices you found in Exercise 5.
3. Solve the following systems of equations using Gaussian elimination. (a) 2x₁ x1 - - 3x2 + 2x3 = 0 x2 + x3 = 7 (b) -x₁ 2 x2 + x3 = 2x + 2x2 - 4.x3 = -4 -x, +5x2+4x3 = 4 x₁ - 2x2 + 3x3 = (c)-x - 3x2 + 2x3 = -2 (d) 2x + 4x2 - 2x3 = 2x + x2 + 3x3 = % 5x+4x2+6x3 + 12 - 2x2 - 4x3 = -2x1 - x2-3x3 = -4 (e) x + x + 4x3 = 4 (f) 2x-3x2 - x3 = 2 2x,+2+3x3 = 5 3.x₁ - 5.x2 5x₁ + 2x2 + 5x3 = 11 = - 2x3 9x1 + 6x2 + 4x3 = -1 4. Solve the problems in Exercise 3 using elimination by pivoting (Gauss-Jordan elimination). 5. (a) Write the LU decomposition for each coefficient matrix A in Ex- ercise 3. (b) Multiply L times U to show that the product is A, for each coef- ficient matrix A in Exercise 3. 6. Find the determinant of each matrix in Exercise 3 using Theorem 2. 7. Re-solve each system in Exercise 3 with the new right-hand-side vector [10, 5, 10] using the numbers in the L and U matrices you found in Exercise 5.
Transcribed Image Text:3. Solve the following systems of equations using Gaussian elimination.
(a) 2x₁
x1
-
-
3x2 + 2x3 = 0
x2 + x3 = 7
(b) -x₁
2
x2 +
x3 =
2x + 2x2
-
4.x3
=
-4
-x, +5x2+4x3 = 4
x₁ - 2x2 + 3x3 =
(c)-x
-
3x2 + 2x3
= -2
(d)
2x + 4x2
-
2x3 =
2x + x2 + 3x3 = %
5x+4x2+6x3 + 12
-
2x2 - 4x3
=
-2x1
-
x2-3x3
= -4
(e) x + x + 4x3 = 4
(f) 2x-3x2
-
x3 =
2
2x,+2+3x3
= 5
3.x₁
-
5.x2
5x₁ + 2x2 + 5x3
= 11
=
-
2x3
9x1 + 6x2 + 4x3
= -1
4. Solve the problems in Exercise 3 using elimination by pivoting
(Gauss-Jordan elimination).
5. (a) Write the LU decomposition for each coefficient matrix A in Ex-
ercise 3.
(b) Multiply L times U to show that the product is A, for each coef-
ficient matrix A in Exercise 3.
6. Find the determinant of each matrix in Exercise 3 using Theorem 2.
7. Re-solve each system in Exercise 3 with the new right-hand-side vector
[10, 5, 10] using the numbers in the L and U matrices you found in
Exercise 5.
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![3. Solve the following systems of equations using Gaussian elimination.
(a) 2x₁
x1
-
-
3x2 + 2x3 = 0
x2 + x3 = 7
(b) -x₁
2
x2 +
x3 =
2x + 2x2
-
4.x3
=
-4
-x, +5x2+4x3 = 4
x₁ - 2x2 + 3x3 =
(c)-x
-
3x2 + 2x3
= -2
(d)
2x + 4x2
-
2x3 =
2x + x2 + 3x3 = %
5x+4x2+6x3 + 12
-
2x2 - 4x3
=
-2x1
-
x2-3x3
= -4
(e) x + x + 4x3 = 4
(f) 2x-3x2
-
x3 =
2
2x,+2+3x3
= 5
3.x₁
-
5.x2
5x₁ + 2x2 + 5x3
= 11
=
-
2x3
9x1 + 6x2 + 4x3
= -1
4. Solve the problems in Exercise 3 using elimination by pivoting
(Gauss-Jordan elimination).
5. (a) Write the LU decomposition for each coefficient matrix A in Ex-
ercise 3.
(b) Multiply L times U to show that the product is A, for each coef-
ficient matrix A in Exercise 3.
6. Find the determinant of each matrix in Exercise 3 using Theorem 2.
7. Re-solve each system in Exercise 3 with the new right-hand-side vector
[10, 5, 10] using the numbers in the L and U matrices you found in
Exercise 5.](https://content.bartleby.com/qna-images/question/3a34a1e4-7c25-4c64-ab7e-d5dd0d192579/0f572f4e-2057-43f2-8778-ecbb12d69ab1/z0ab0w_processed.png)
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