Faded examples · Describing the transformation a matrix represents

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Each example shows a little less than the one before; complete the faded (blank) steps yourself, using the same method every time: find the image of $(1,0)$ (the first column) and of $(0,1)$ (the second column), then name the transformation fully. Also answer the check question. No calculator.

①Example 1fully worked: read it through
Describe fully $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$.
1Image of $(1,0)$ = first column
$(1,0)\to(1,0)$
2Image of $(0,1)$ = second column
$(0,1)\to(0,-1)$
3Name it fully (line / direction / centre)
reflection in the $x$-axis
Check · Why is this a reflection in the $x$-axis (not the $y$-axis)?
A points on the $x$-axis stay fixed and the $y$-coordinate changes signB both coordinates change signC the matrix is the identityD it doubles every point
②Example 2you finish the last 1 step
Describe fully $\begin{pmatrix}0&1\\1&0\end{pmatrix}$.
1Image of $(1,0)$ = first column
$(1,0)\to(0,1)$
2Image of $(0,1)$ = second column
$(0,1)\to(1,0)$
3Name it fully (line / direction / centre)
reflection in the line $y=x$
Check · A student said this was a rotation. What tells you it is a reflection?
A the coordinates simply swap; points on $y=x$ are unchangedB the determinant is 2C the matrix is not squareD nothing, it is a rotation
③Example 3you finish the last 2 steps
Describe fully $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$.
1Image of $(1,0)$ = first column
$(1,0)\to(0,1)$
2Image of $(0,1)$ = second column
$(0,1)\to(-1,0)$
3Name it fully (line / direction / centre)
rotation $90^{\circ}$ anticlockwise about the origin
Check · How do you know the rotation is anticlockwise, not clockwise?
A $(1,0)$ turns to $(0,1)$, which is a quarter-turn anticlockwiseB the numbers are negativeC all rotations are anticlockwiseD because the centre is the origin
④Example 4your turn: every step
Describe fully $\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$.
1Image of $(1,0)$ = first column
$(1,0)\to(-1,0)$
2Image of $(0,1)$ = second column
$(0,1)\to(0,-1)$
3Name it fully (line / direction / centre)
rotation $180^{\circ}$ about the origin
Check · Why can you also NOT call this an enlargement?
A a $180^{\circ}$ rotation keeps every length and area the same; an enlargement would change sizeB the matrix has negative entriesC enlargements are always positiveD it maps the origin to itself

Answers · Matrix Transformations

Faded examples · Describing the transformation a matrix represents
① Example 1   $(1,0)\to(1,0)$→$(0,1)\to(0,-1)$→reflection in the $x$-axis   $\text{Reflection in the }x\text{-axis}$
Check: A: points on the $x$-axis stay fixed and the $y$-coordinate changes sign
② Example 2   $(1,0)\to(0,1)$→$(0,1)\to(1,0)$→reflection in the line $y=x$   $\text{Reflection in the line }y=x$
Check: A: the coordinates simply swap; points on $y=x$ are unchanged
③ Example 3   $(1,0)\to(0,1)$→$(0,1)\to(-1,0)$→rotation $90^{\circ}$ anticlockwise about the origin   $\text{Rotation }90^{\circ}\text{ anticlockwise about the origin}$
Check: A: $(1,0)$ turns to $(0,1)$, which is a quarter-turn anticlockwise
④ Example 4   $(1,0)\to(-1,0)$→$(0,1)\to(0,-1)$→rotation $180^{\circ}$ about the origin   $\text{Rotation }180^{\circ}\text{ about the origin}$
Check: A: a $180^{\circ}$ rotation keeps every length and area the same; an enlargement would change size
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