Which 3 x 3 matrix represents a reflection in the yz plane followed by inversion through the origin?
That would be the product of the matrices of the two operations, done in that order. The first matrix changes x tp -x, and leaves y and z the same. The second matrix changes x to -x, y to -y and z to -z.
The product matrix leaves x unchanged and changes y to -y and z to -z.
The resulting matrix would seem to be
1...0...0
0.. -1..0
0...0..-1
To find the 3x3 matrix representing a reflection in the yz plane followed by inversion through the origin, we need to understand the operations involved.
A reflection in the yz plane is a transformation that flips the coordinates of a point across the yz plane. This means that the x-coordinate remains the same, while the signs of the y and z coordinates are reversed.
On the other hand, inversion through the origin is a transformation that maps each point to its antipodal point on the sphere centered at the origin. Inverting a point through the origin involves multiplying each coordinate by -1.
Now, let's set up a matrix with the elements that represent these operations:
| x' | | a b c | | x |
| y' | = | d e f | ⋅ | y |
| z' | | g h i | | z |
To reflect in the yz plane, the x-coordinate remains unchanged (x' = x), while the sign of y and z is reversed:
| x | | 1 0 0 | | x |
| y | = | 0 -1 0 | ⋅ | y |
| z | | 0 0 -1 | | z |
To invert through the origin, each coordinate is multiplied by -1:
| x | | -1 0 0 | | x |
| y | = | 0 -1 0 | ⋅ | y |
| z | | 0 0 -1 | | z |
Combining the two operations, we multiply these matrices together:
| x' | | -1 0 0 | | 1 0 0 | | x |
| y' | = | 0 -1 0 | ⋅ | 0 -1 0 | ⋅ | y |
| z' | | 0 0 -1 | | 0 0 -1 | | z |
Simplifying this matrix multiplication, we get:
| x' | | -1 0 0 | | x |
| y' | = | 0 1 0 | ⋅ | y |
| z' | | 0 0 1 | | z |
Therefore, the 3x3 matrix representing a reflection in the yz plane followed by inversion through the origin is:
| -1 0 0 |
| 0 1 0 |
| 0 0 1 |