Find the entries of the following matrices:

(a) the 2 ×2 matrix M for the reflection across the line y = x.
(b) the 2 ×2 matrix N for the 90 degree counterclockwise rotation about the origin.
(c) the product MN;what transformation does this represent?
(d) the product NM; what transformation does this represent?
(e) the product MM; what transformation does this represent?

i hate exeter too

no we literally still have to do it. still Exeters problem sadly

atleast exeter got rid of this. now it's peddie's problem

(a) The 2×2 matrix M for the reflection across the line y = x is [[0, 1], [1, 0]].

(b) The 2×2 matrix N for the 90 degree counterclockwise rotation about the origin is [[0, -1], [1, 0]].

(c) The product MN represents a reflection followed by a rotation, resulting in a transformation known as a "reflected rotation" or a "compound transformation".

(d) The product NM represents a rotation followed by a reflection, resulting in a transformation known as a "rotated reflection" or a "compound transformation".

(e) The product MM represents a reflection followed by another reflection, resulting in a transformation known as a "double reflection" or a "compound transformation".

To find the entries of the matrices and determine the transformations, we need to understand the basic operations and properties of matrices.

(a) Reflection across the line y = x:
To reflect a point across a line, the x and y coordinates of the point are swapped. So, to find M, we replace each entry in the matrix with its corresponding reflection:
M = [[0, 1], [1, 0]]

(b) 90-degree counterclockwise rotation about the origin:
To rotate a point 90 degrees counterclockwise about the origin, we use the rotation matrix:
N = [[0, -1], [1, 0]]

(c) Product MN:
To find the product of two matrices, we multiply the corresponding entries of the matrices. So, multiply the matrices M and N:
MN = [[0, -1], [1, 0]] * [[0, 1], [1, 0]]

Performing the matrix multiplication, we get:
MN = [[-1, 0], [0, 1]]

The transformation represented by the product MN is a reflection across the y-axis.

(d) Product NM:
To find the product of matrices N and M:
NM = [[0, 1], [1, 0]] * [[0, -1], [1, 0]]

Performing the matrix multiplication:
NM = [[1, 0], [0, -1]]

The transformation represented by the product NM is a reflection across the x-axis.

(e) Product MM:
To find the product of matrices M and M, multiply the matrices:
MM = [[0, 1], [1, 0]] * [[0, 1], [1, 0]]

Performing the matrix multiplication:
MM = [[1, 0], [0, 1]]

The transformation represented by the product MM is an identity transformation, which means it does not change the position of the points.

In summary:
(a) Matrix M represents reflection across the line y = x.
(b) Matrix N represents a 90-degree counterclockwise rotation about the origin.
(c) The product MN represents a reflection across the y-axis.
(d) The product NM represents a reflection across the x-axis.
(e) The product MM represents the identity transformation.