Chapter 8, Problem 1PS

(a)

To determine

The value of AT and AAT.

Answer to Problem 1PS

The values are,

  $AT=\left[\begin{array}{l}\begin{array}{ccc}-1& -4& -2\end{array}\\ \begin{array}{ccc}1& 2& 3\end{array}\end{array}\right]$ ; $AAT=\left[\begin{array}{l}\begin{array}{ccc}-1& -2& -3\end{array}\\ \begin{array}{ccc}-1& -4& -2\end{array}\end{array}\right]$

Explanation of Solution

Given information:

The matrix T shows the coordinates of the vertices of a triangle:

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

The transformation matrix is:

  $A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

Formula used:

Matrix multiplication is used.

Calculation:

Order of matrix A is $2\times 2$

Order of matrix T is $2\times 3$

Since the number of columns in matrix A equals the number of rows in matrix T, so the product AT is defined.

  $\begin{array}{l}AT=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{l}\begin{array}{ccc}1& 2& 3\end{array}\\ \begin{array}{ccc}1& 4& 2\end{array}\end{array}\right]\\ =\left[\begin{array}{l}\begin{array}{ccc}0\left(1\right)-1\left(1\right)& 0\left(2\right)-1\left(4\right)& 0\left(3\right)-1\left(2\right)\end{array}\\ \begin{array}{ccc}1\left(1\right)+0\left(1\right)& 1\left(2\right)+0\left(4\right)& 1\left(3\right)+0\left(2\right)\end{array}\end{array}\right]\\ AT=\left[\begin{array}{l}\begin{array}{ccc}-1& -4& -2\end{array}\\ \begin{array}{ccc}1& 2& 3\end{array}\end{array}\right]\end{array}$

Order of matrix is $2\times 2$

Order of matrix AT is $2\times 3$

Since the number of columns in matrix A equals the number of rows in matrix AT, so the product A (AT) is defined.

  $\begin{array}{l}AAT=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{l}\begin{array}{ccc}-1& -4& -2\end{array}\\ \begin{array}{ccc}1& 2& 3\end{array}\end{array}\right]\\ =\left[\begin{array}{l}\begin{array}{ccc}0\left(-1\right)-1\left(1\right)& 0\left(-4\right)-1\left(2\right)& 0\left(-2\right)-1\left(3\right)\end{array}\\ \begin{array}{ccc}1\left(-1\right)+0\left(1\right)& 1\left(-4\right)+0\left(2\right)& 1\left(-2\right)+0\left(3\right)\end{array}\end{array}\right]\\ AAT=\left[\begin{array}{l}\begin{array}{ccc}-1& -2& -3\end{array}\\ \begin{array}{ccc}-1& -4& -2\end{array}\end{array}\right]\end{array}$

The triangle is as follows

  

Conclusion:

The values are,

  $AT=\left[\begin{array}{l}\begin{array}{ccc}-1& -4& -2\end{array}\\ \begin{array}{ccc}1& 2& 3\end{array}\end{array}\right]$ ; $AAT=\left[\begin{array}{l}\begin{array}{ccc}-1& -2& -3\end{array}\\ \begin{array}{ccc}-1& -4& -2\end{array}\end{array}\right]$

(b)

To determine

The transformation produced by the triangle that is determined by AT and T.

Answer to Problem 1PS

The vertices are (1,1) (3,2) (2,4)

Explanation of Solution

Given information:

The matrix T shows the coordinates of the vertices of a triangle:

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

The transformation matrix is:

  $A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

Formula used:

Matrix multiplication is used.

Calculation:

The triangle is as follows

  

Conclusion:

The vertices are (1,1) (3,2) (2,4)