10.9. EXERCISES 219

18. Find a PLU factorization of

1 2 12 4 11 0 22 2 1

 and use it to solve the systems

a.

1 2 12 4 11 0 22 2 1

 x

yz

=

1211

 b.

1 2 12 4 11 0 22 2 1

 x

yz

=

abcd



19. Find a PLU factorization of

 0 2 1 22 1 −2 02 3 −1 2

 and use it to solve the systems

a.

 0 2 1 22 1 −2 02 3 −1 2



xyzw

=

 112



b.

 0 2 1 22 1 −2 02 3 −1 2



xyzw

=

 213

20. Find a QR factorization for the matrix 1 2 1

3 −2 11 0 2

21. Find a QR factorization for the matrix 1 2 1 0

3 0 1 11 0 2 1

22. If you had a QR factorization, A = QR, describe how you could use it to solve the

equation Ax = b. This is not usually the way people solve this equation. However,the QR factorization is of great importance in certain other problems, especially infinding eigenvalues and eigenvectors.

23. In this problem, is another explanation of the LU factorization. Let

A =

 1 2 32 0 −21 3 1

