Chapter 4.2, Problem 34PPS

(a)

To determine

To Find: The matrix to represent $\Delta ABC$ .

Answer to Problem 34PPS

The required matrix is $\left[\begin{array}{ccc}4& -1& -3\\ 2& -1& 1\end{array}\right]$ .

Explanation of Solution

Given:

The given graph is shown in Figure 1

  

Figure 1

The given matrix is,

  $\left[\begin{array}{ccc}-3& -4& 1\\ 8& 6& 0\end{array}\right]$

The vertices of the triangle is $\left(-3,8\right),\left(-4,6\right)\left(1,0\right)$ .

Calculation:

Consider the vertices from figure 1 is,

  $A\left(4,2\right),B\left(-1,1\right),C\left(-3,1\right)$

Then, the matrix of the triangle is,

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

(b)

To determine

To Find: The matrix to represent $\Delta ABC$ .

Answer to Problem 34PPS

The required matrix is $\left[\begin{array}{ccc}8& -2& -6\\ 4& -2& 2\end{array}\right]$ and graph for it is shown in Figure 2.

Explanation of Solution

Consider the given matrix is,

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

Then, multiply it with scalar 2 as,

  $2\left[\begin{array}{ccc}4& -1& -3\\ 2& -1& 1\end{array}\right]=\left[\begin{array}{ccc}8& -2& -6\\ 4& -2& 2\end{array}\right]$

The graph for the new matrix is shown in Figure 2

  

Figure 2

(c)

To determine

To Find: The conjecture for the result of multiplying the matrix by 0.5.

Answer to Problem 34PPS

The sides is doubled in case of scalar 2 and halved in case of scalar 0.5.

Explanation of Solution

Consider when the original matrix is multiplied by 2 then the triangle is of same shape but sides doubled.

But, when the matrix is multiplied by 0.5 the triangle is similar and reduce by half on every side.