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A graphics program uses the matrix T to transform a triangle with vertices (0, 0), (2, 0) and (0…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A graphics program uses the matrix $T$ to transform a triangle with vertices $(0, 0)$ , $(2, 0)$ and $(0, 3)$ .

\displaystyle T = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}

Find the area before, the determinant of $T$ and the area after. Then find the area after a second matrix $S$ , and say whether $S$ flips the picture.

\displaystyle S = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}

Need a reminder? The key ideaShow

The area of the triangle with vertices $(x_i, y_i)$ is $\tfrac{1}{2}\left|\det\right|$ of the matrix with rows $(x_i, y_i, 1)$ . Three points are collinear exactly when this determinant is $0$ . A matrix multiplies every area by $|\det T|$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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