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I am stuck up with this question from my Linear Algebra Assignment which states to explain geometrically why

$$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix}-1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$ I don't know where to start with. Could anyone kindly help me with this?

Archaick
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Vishal K Nair
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2 Answers2

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The first matrix is a reflection in the x-axis $(x, y) \to (x, -y)$,

the second one is a reflection in the y-axis $(x, y) \to (-x, y)$

it doesn't matter which order you perform reflections about mutually perpendicular axes.

The point $(x,y)$ will be mapped to $(-x, -y)$ in either case.

WW1
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  • it will be very kind of you if you could explain how first matrix is a reflection in the x-axis y→ −y ? – Vishal K Nair May 17 '15 at 04:39
  • the x-axis is a horizontal line that divides the upper half plane from the lower half plane. These two half planes are reflections of each other when you reflex in the x-axis, so it is the $y$ co-ordinates that changes sign. – WW1 May 17 '15 at 05:03
  • oh yes ....! I got that, thanks a lot ! – Vishal K Nair May 17 '15 at 05:11
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Note that these matrices are reflections and that your equation is equivalent to $$\text{Ref}_{x=0}\circ \text{Ref}_{y=0}=\text{Ref}_{y=0}\circ \text{Ref}_{x=0}$$ where $\text{Ref}_{f}$ is the transformation which reflects the plane through the line $f$. But reflection through $y=0$ and then $x=0$ is the same as reflection through $x=0$ and then $y=0$. Hope that helps!

Archaick
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  • It depends upon if the OP knows about linear transformations –  May 17 '15 at 04:29
  • @Rememberme OP's going to have to know how to interpret a matrix geometrically one way or another to do this problem. – anon May 17 '15 at 04:30
  • @Rememberme I will be happy to go into more detail regarding how we can see that these matrices represent the reflections I claim if he/she asks. – Archaick May 17 '15 at 04:31
  • this only works on mirrors at right angles. it is so here. – abel May 17 '15 at 04:39