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A graphics program moves every point (x, y) of a picture to the point (2x + y, x + y)…

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

A graphics program moves every point $(x, y)$ of a picture to the point $(2x + y,\ x + y)$ . After the move, one corner of the picture is at $(7, 4)$ . Find the original position of this corner by writing the move as $AX = B$ and solving with $A^{-1}$ .

\displaystyle \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}

$. The new position is$ B = (7, 4)$.

\displaystyle \begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}

Need a reminder? The key ideaShow

$AX = B$ has the unique solution $X = A^{-1}B$ when $|A| \ne 0$ . When $|A| = 0$ there is no solution or infinitely many. A homogeneous system $AX = O$ has a non-zero solution exactly when $|A| = 0$ .

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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