Linear Algebra with Applications, 10th Edition

Section 7.2 Exercises

1. Let

   $$A=\left[\begin{array}{ccc}\hfill 1& 1& \hfill 1\\ \hfill 2& 4& \hfill 1\\ \hfill -3& 1& \hfill -1\end{array}\right]$$

   Factor A into a product LU, where L is lower triangular with 1’s along the diagonal and U is upper triangular.

2. Let A be the matrix in Exercise 1. Use the LU factorization of A to solve $A\mathbf{x}=\mathbf{b}$ for each of the following choices of b:

   1. ${(4,3,-13)}^T$

   2. ${(3,1,-10)}^T$

   3. ${(7,23,0)}^T$

3. Let A and B be $n\times n$ matrices and let $\mathbf{x}\in {ℝ}^n$.

   1. How many scalar additions and multiplications are necessary to compute the product Ax?

   2. How many scalar additions and multiplications are necessary to compute the product AB?

   3. How many scalar additions and multiplications are necessary to compute (AB)x? To compute A(Bx)?

4. Let $A\in {ℝ}^{m\times n},B\in {ℝ}^{n\times r}$, and $\mathbf{\text{x}},\mathbf{\text{y}}\in {\mathbb{\text{R}}}^n$. Suppose that the product $A{\mathbf{\text{xy}}}^{T}B$ is computed in the following ways:

   1. $\left(A\left({\mathbf{\text{xy}}}^T\right)\right)B$

   2. $\left(A\mathbf{x}\right)\left({\mathbf{y}}^{T}B\right)$

   3. $((A\mathbf{x}$