218 CHAPTER 10. A FEW FACTORIZATIONS

10. Find the LU factorization of the coefficient matrix using Dolittle’s method and use itto solve the system of equations.

x+2y = 52x+3y = 6

11. Find the LU factorization of the coefficient matrix using Dolittle’s method and use itto solve the system of equations.

x+2y+ z = 1y+3z = 2

2x+3y = 6

12. Find the LU factorization of the coefficient matrix using Dolittle’s method and use itto solve the system of equations.

x+2y+3z = 52x+3y+ z = 6x− y+ z = 2

13. Find the LU factorization of the coefficient matrix using Dolittle’s method and use itto solve the system of equations.

x+2y+3z = 52x+3y+ z = 6

3x+5y+4z = 11

14. Is there only one LU factorization for a given matrix? Hint: Consider the equation(0 10 1

)=

(1 01 1

)(0 10 0

).

Look for all possible LU factorizations.

15. Find a PLU factorization of

 1 2 11 2 22 1 1

 .

16. Find a PLU factorization of

 1 2 1 2 12 4 2 4 11 2 1 3 2

 .

17. Find a PLU factorization of

1 2 11 2 22 4 13 2 1

 .