Linear Algebra with Applications, 10th Edition

7.2

1. 1. $A=\left[\begin{array}{ccc}\hfill 1& 0& 0\\ \hfill 2& 1& 0\\ \hfill -3& 2& 1\end{array}\right]\left[\begin{array}{ccc}1& 1& \hfill 1\\ 0& 2& \hfill -1\\ 0& 0& \hfill 3\end{array}\right]$

2. 2. 

   1. (a) ${(2,-1,3)}^T$;

   2. (b) ${(1,-1,3)}^T$;

   3. (c) ${(1,5,1)}^T$

3. 3. 

   1. (a) $n^2$ multiplications and $n(n-1)$ additions;

   2. (b) $n^3$ multiplications and $n^2(n-1)$ additions;

   3. (c) (AB)x requires $n^3+n^2$ multiplications and $n^3-n$ additions; A(Bx) requires $2n^2$ multiplications and $2n(n-1)$ additions.

4. 4. 

   1. (b)  

      1. 156 multiplications and 105 additions,

      2. 47 multiplications and 24 additions,

      3. 100 multiplications and 60 additions

5. 8. $5n-4$ multiplications/divisions, $3n-3$ additions/subtractions

6. 9. 

   1. (a) $\left[(n-j)(n-j+1)\right]/2$ multiplications; $\left[(n-j-1)(n-j)\right]/2$ additions;

   2. (c) It requires on the order of $\frac{2}{3}{n}^3$ additional multiplications/divisions to compute $A^{-1}$ given the LU factorization.

7. 10. Hint: The operation count for the multiplication $A^{-1}\mathbf{b}$ is the same ...