Chapter 9.5, Problem 61PE

Given a square matrix $A$ . elementary row operations (or column operations) performed on $A$ affect the value of $\left|A\right|$ , in the following ways:

• Interchanging any two rows (or columns) of $A$ will change the sign of $\left|A\right|$

• Multiplying a row (or column) of $A$ by a constant real number $k$ multiplies $\left|A\right|$ by $k$ .

• Adding a multiple of a row (or column) of $A$ to another row (or column) of $A$ does not change the value of $\left|A\right|$ .

For Exercises 59-64, demonstrate these three properties

Given $A=\left[\begin{array}{cc}1& -3\\ 4& 1\end{array}\right]\text{ and }B=\left[\begin{array}{cc}2& -6\\ 4& 1\end{array}\right]$ ,

a. Evaluate $\left|A\right|$ .

b. Evaluate $\left|B\right|$

c. How are $A$ and $B$ related and how are $\left|A\right|$ and $\left|B\right|$ related?