Question

Use matrix multiplication to find the image of the triangle with data matrix \(D = \left[ {\begin{array}{*{20}{c}}{\bf{5}}&{\bf{2}}&{\bf{4}}\\{\bf{0}}&{\bf{2}}&{\bf{3}}\end{array}} \right]\) under the transformation that reflects points through the y-axis. Sketch both the original triangle and its image.

Short answer

\(\left[ {\begin{array}{*{20}{c}}{ - 5}&{ - 2}&{ - 4}\\0&2&3\end{array}} \right]\)

Step-by-step solution

Find the transformed matrix

The matrix of transformation is \(A = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&1\end{array}} \right]\). The transformed data matrix is shown below:

\(\begin{array}{c}AD = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5&2&4\\0&2&3\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ - 5}&{ - 2}&{ - 4}\\0&2&3\end{array}} \right]\end{array}\)

Sketch the original triangle and the transformed triangle

The figure below represents the transformed triangle.

So, the transformed matrix is \(\left[ {\begin{array}{*{20}{c}}{ - 5}&{ - 2}&{ - 4}\\0&2&3\end{array}} \right]\).