MA2330: Reading Assignment for 1 March

1.

Read Sections 3.3.

2.

Make sure that you understand:

• How to use Cramer's rule to solve a linear system.
• That Cramer's rule is not computationally feasible unless the size of the system is very small. For an $n\times n$ system where nis even moderately large, Cramer's rule is extremely inefficient, and should be regarded as a theoretical tool. (Many people, though, including myself, like to use Cramer's rule for solving a $2\times 2$system because no division is necessary until the last step, an advantage if the original matrix has integer entries.)
• The formula for A^-1. Like Cramer's rule, this is a theoretical tool; it is not computationally efficient.
• The geometric meaning of the determinant.

3.

Questions (to be answered by email at least one hour before class):

(a)

Suppose A is an $n\times n$ matrix, and we need to find x₁, the first component of the solution to $A{\bf x}={\bf b}$ (assume that we do not need the other components).

i.

About how many arithmetic operations (in terms of n) would it take to find x₁ using Cramer's rule?

ii.

Is is possible to find x₁ using row operations without computing all of ${\bf x}$?

iii.

About how many arithmetic operations would it take to compute x₁ using row operations?

(b)

Let S be the region (in the plane) enclosed by the rectangle with vertices (-1,0), (2,0), (2,2), (-1,2), and let

$$A=\left[\begin{array}{rr}2&3\\ 0&4\end{array}\right].$$

Define

$$T=\{Ax\,:\,x\in S\}.$$

What is the area of T? How did you find it?