Question

How do you multiply `((-1, 0), (0, -1))` with `((-1, -3, -3, 2), (3, -1, -2, 1))`?

Answer 1

`((1,3,3,-2),(-3,1,2,-1))`

Explanation:

Remember multiply row by column:
#( (color(red)(-1),color(red)(0)),(color(blue)(0),color(blue)(-1))) xx ((-1,-3,-3,2),(3,-1,-2,1) )#

So the product matrix will be composed of:

`color(white)("XX")`Row 1
`color(white)("XXXXX")`Column 1: `color(red)(-1)xx(-1)+color(red)(0)xx3 = 1`
`color(white)("XXXXX")`Column 2: `color(red)(-1)xx(-3)+color(red)(0)xx(-1) = 3`
`color(white)("XXXXX")`Column 3: `color(red)(-1)xx(-3)+color(red)(0)xx(-2) = 3`
#color(white) ("XXXXX")#Column 4: `color(red)(-1)xx2+color(red)(0)xx1 = -2`

`color(white)("XX")`Row 2
`color(white)("XXXXX")`Column 1: `color(blue)0xx(-1)+color(blue)(""(-1))xx3 = -3`
`color(white)("XXXXX")`Column 2: `color(blue)0xx(-3)+color(blue)(""(-1))xx(-1) = 1`
`color(white)("XXXXX")`Column 3: `color(blue)0xx(-3)+color(blue)(""(-1))xx(-2) = 2`
`color(white)("XXXXX")`Column 4: `color(blue)0xx2+color(blue)(""(-1))xx1 = -1`

`=((1,3,3,-2),(-3,1,2,-1))`

Answer 2

`((-1, 0), (0, -1))` has the effect of changing the signs of the other matrix.

Explanation:

`((-1, 0), (0, -1))` is one of the forms of the identity or unit matrix.

The positioning of the 0's and 1's in this matrix means that in multiplying with this matrix, the numbers in the other matrix stay the same, but the negative signs have the effect of changing the signs of the other matrix.

This is used as one of the transformation matrices and gives the co-ordinates of the image under the following transformations:

A rotation of 180° about the centre at the origin, (0,0)

A dilation (or enlargement) with SF = -1 and the centre at (0.0)

Once you recognise what this is, you will have the satisfaction of being able to give the answer immediately.

Check for yourself that this actually works.