Q37. - Algebra 2

Question

For Exercises 34-38, use the figure below.

37. Make a conjecture about what transformation ${\mathbf{B}}^{-1}$describes on a coordinate plane.

Step-by-Step Solution

Answer

The conjecture about the transformation${\mathbf{B}}^{-1}$ describes on a coordinate plane is that the transformed rectangle will be half times reduced that implies all the linear measures of the transformed rectangle will be half times the linear measures of the given rectangle.

Step 1 ­- Observe the given figure.

The given figure is:

Step 2 ­- Description of step.

It is being given that $B=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

Therefore, the value of the determinant of matrixB is:

$\begin{array}{c}\left|\begin{array}{cc}2& 0\\ 0& 2\end{array}\right|=\left(2\right)\left(2\right)-\left(0\right)\left(0\right)\\ =4-0\\ =4\end{array}$

The inverse of$2\times 2$ matrix$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is given by$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ and $ad-bc\ne 0$.

Therefore, the inverse of matrixB which is$B^{-1}$ is given by:

$\begin{array}{c}{B}^{-1}=\frac{1}{\left(2\right)\left(2\right)-\left(0\right)\left(0\right)}\left[\begin{array}{cc}2& -\left(0\right)\\ -\left(0\right)& 2\end{array}\right]\\ =\frac{1}{4-0}\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]\\ =\frac{1}{4}\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]\\ =\left[\begin{array}{cc}\frac{2}{4}& \frac{0}{4}\\ \frac{0}{4}& \frac{2}{4}\end{array}\right]\\ =\left[\begin{array}{cc}\frac{1}{2}& 0\\ 0& \frac{1}{2}\end{array}\right]\end{array}$

Step 3 ­- Write a conjecture about what transformation B - 1 describes on a coordinate plane.

It can be noticed that the matrix B is$\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$ and the transformation was that the transformed rectangle was enlarged two times that implies all the linear measures of the transformed rectangle were two times the linear measures of the given rectangle.

As the matrix$B^{-1}$ is$\left[\begin{array}{cc}\frac{1}{2}& 0\\ 0& \frac{1}{2}\end{array}\right]$ , therefore the transformation after multiplying the matrix$B^{-1}$ with vertex matrix of the given rectangle is that the transformed rectangle will be half times reduced that implies all the linear measures of the transformed rectangle will be half times the linear measures of the given rectangle.

Therefore, the conjecture about the transformation$B^{-1}$ describes on a coordinate plane is that the transformed rectangle will be half times reduced that implies all the linear measures of the transformed rectangle will be half times the linear measures of the given rectangle.