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Tutorial 7 mth 3201
- 3. 1(a)
3/7 β6/7 2/7
π΄ π = 2/7 3/7 6/7
6/7 2/7 β3/7
3/7 2/7 6/7 3/7 β6/7 2/7 1 0 0
π
π΄π΄ = β6/7 3/7 2/7 2/7 3/7 6/7 = 0 1 0 = πΌ
2/7 6/7 β3/7 6/7 2/7 β3/7 0 0 1
π΄π΄ π = πΌ
π΄π΄ π = π΄π΄β1
β΄ π΄ π = π΄β1
Since π΄ π = π΄β1 , π΄ is an orthogonal matrix
- 4. 1(b)
π
ππ€ π ππππ ππ π΄: π1 = 3/7 2/7 6/7
π2 = β6/7 3/7 2/7
π3 = 2/7 6/7 β3/7
π1 , π2 = 3/7 β6/7 + (2/7)(3/7) + 6/7 2/7 = 0
π1 , π3 = 3/7 2/7 + (2/7)(6/7) + 6/7 β3/7 = 0
π2 , π3 = β6/7 2/7 + (3/7)(6/7) + 2/7 β3/7 = 0
π1 = π1 , π1 1/2 = 3/7 2 + 2/7 2 + 6/7 2 = 1
π2 = π2 , π2 1/2 = β6/7 2 + 3/7 2 + 2/7 2 = 1
π3 = π3 , π3 1/2 = 2/7 2 + 6/7 2 + β3/7 2 =1
β΄ r1 , r2 , r3 is an orthonormal set. Hence, it is an orthogonal matrix
- 5. 1(c)
3/7
πΆπππ’ππ π ππππ ππ π΄:π1 = β6/7 ,
2/7
2/7 6/7
π2 = 3/7 , π3 = 2/7
6/7 β3/7
π1 , π2 = 3/7 2/7 + (β6/7)(3/7) + 2/7 6/7 = 0
π1 , π3 = 3/7 6/7 + (β6/7)(2/7) + 2/7 β3/7 = 0
π2 , π3 = 2/7 6/7 + (3/7)(2/7) + 6/7 β3/7 = 0
π1 = π1 , π1 1/2 = 1
π2 = π2 , π2 1/2 = 1
π3 = π3 , π3 1/2 =1
β΄ c1 , c2 , c3 is an orthonormal set. Hence, it is an orthogonal matrix
- 6. 2. Inverse of matrix A
π΄ πΌ πΌ π΄β1
3/7 2/7 6/7 1 0 0 π 7π
1 , 7π
2 , 7π
3
β6/7 3/7 2/7 0 1 0 ππ π
1 β π
2 , 2π
1 + π
2 , π
2 + 3π
3
2/7 6/7 β3/7 0 0 1 π
2
πππ , 3π
2 β π
3
7
π
3
ππ£
49
π£ π
2 β 2π
3
π£π π
1 + 4π
2 β 9π
3
1 0 0 3/7 β6/7 2/7
0 1 0 2/7 3/7 6/7
0 0 1 6/7 2/7 β3/7
β΄ A is an orthogonal matrix from question 1
π΄ π = π΄β1
- 7. 3(a) Transition matrix B to Bβ²
1 2 2 3
ππ΅ = , π π΅β² =
3 2 1 β4
π π΅β² π π΅ πΌ π
2 3 1 2 π π
1 β π
2 ππ 2π
1 β π
2 1 0 13/11 14/11
1 β4 3 2 βπ
2 0 1 β5/11 β2/11
πππ ππ£ π
1 + 4π
2
11 13/11 14/11
β΄ Transition matrix , P =
β5/11 β2/11
3(b) Transition matrix Bβ² to B
π π΅ π π΅β² πΌ π βπ
2
π π
2 β 3π
1 ππ
1 2 2 3 4 1 0 β1/2 β7/2
3 2 1 β4 πππ π
1 β 2π
2 0 1 5/4 13/4
β1/2 β7/2
β΄ Transition matrix , Q =
5/4 13/4
- 8. β1 From 3(a)
3(c) Find π€ π΅β² ππ π€ π΅ =
3 π π΅β² π π΅ πΌ π
using equation π£ = π π£ 13/11 14/11
π΅β² π΅ Transition matrix , P =
β5/11 β2/11
π€ π΅β² = π π€ π΅
13/11 14/11 β1
π€ π΅β² =
β5/11 β2/11 3
29/11
π€ π΅β² =
β1/11
- 9. 4(a) Transition matrix B to Bβ²
1 β1 β1 3 0 1
ππ΅ = 2 β1 1 , π π΅β² = 7 4 0 π π΅β² π π΅ πΌ π
1 1 7 β2 1 0
3 0 1 1 β1 β1 ERO 1 0 0 β2/15 β1/3 β9/5
7 4 0 2 β1 1 0 1 0 11/15 1/3 17/5
β2 1 0 1 1 7 0 0 1 7/5 0 22/5
β΄ Transition matrix , P
4(b) Transition matrix Bβ² to B
π π΅ π π΅β² πΌ π
1 β1 β1 3 0 1 ERO 1 0 0 11 11 β4
2 β1 1 7 4 0 0 1 0 23/2 29/2 β13/2
1 1 7 β2 1 0 0 0 1 β7/2 β7/2 3/2
11 11 β4
β΄ Transition matrix , Q = 23/2 29/2 β13/2
β7/2 β7/2 3/2
- 10. From 4(a)
4(c) Find w Bβ² first and then find w B
π π΅β² π π΅ πΌ π
π π΅β² π€ πΌ w Bβ² 13/11 14/11
Transition matrix , P =
3 0 1 β2 1 0 0 β22/15 β5/11 β2/11
ERO
7 4 0 β6 0 1 0 16/15
β2 1 0 4 0 0 1 12/5
w Bβ²
using equation π£ π΅ = πβ1 π£ π΅β²
π€ π΅ = πβ1 π€ π΅β²
π€= π π€ π΅β²
π΅
11 11 β4 β22/15
π€ π΅ = 23/2 29/2 β13/2 16/15
β7/2 β7/2 3/2 12/5
β14
π€ π΅ = β17
5
- 11. 4(d) Find w B directly
ππ΅ π€ πΌ w B
1 β1 β1 β2 β2π
1 + π
2 1 β1 β1 β2 π
1 + π
2 1 0 2 β4
2 β1 1 β6 0 1 3 β2 0 1 3 β2
βπ
1 + π
3 β2π
2 + π
3
1 1 7 4 0 2 8 6 0 0 2 10
π
3 /2 1 0 2 β4 π
1 β 2π
2 1 0 0 β14
0 1 3 β2 π
2 β 3π
3 0 1 0 β17
0 0 1 5 0 0 1 5
β14
β΄ π€ π΅ = β17
5
- 12. 5(a) Transition matrix B to Bβ²
3 β4 2 1
ππ΅ = , π π΅β² =
1 2 0 3
π π΅β² π π΅ πΌ π
2 1 3 β4
0 3 1 2
π π
1 /2
ππ π
2 /3
π
2
πππ π
1 β
2
1 0 4/3 β7/3
0 1 1/3 2/3
4/3 β7/3
β΄ Transition matrix , P =
1/3 2/3
- 13. 5(b) Find q B first and then find q Bβ²
ππ΅ π πΌ q B
3 β4 β1 ERO 1 0 7/5
1 2 4 0 1 13/10
q B
using equation π£ π΅β² = π π£ π΅
π = π π π΅
π΅β²
4/3 β7/3 7/5
π π΅β² =
1/3 2/3 13/10
β7/6
π π΅β² =
4/3
- 14. 5(c) Find q Bβ² directly
π π΅β² π πΌ q Bβ²
2 1 β1 π
1 /2
1 1/2 β1/2
0 3 4 π
2 /3 0 1 4/3
βπ
2
+ π
1
2 1 0 β7/6
0 1 4/3
β7/6
β΄ π π΅β² =
4/3