Let A = (aij) be the 2x2 matrix of reflection in the x-axis followed by reflection in the line y = x.Find a12

To find the matrix A that represents the reflection in the x-axis followed by reflection in the line y = x, we first need to find the matrices that represent each of these transformations individually.

1. Reflection in the x-axis: This transformation flips the sign of the y-coordinate, while the x-coordinate remains the same. The matrix that represents this transformation is:

   Rx = [1 0] [0 -1]

2. Reflection in the line y = x: This transformation swaps the x and y coordinates. The matrix that represents this transformation is:

   Ryx = [0 1] [1 0]

Now, to find the matrix A that represents the combined transformation, we multiply these two matrices together. Remember that matrix multiplication is not commutative, so the order in which we multiply the matrices matters. Since the reflection in the x-axis happens first, we multiply Rx by Ryx:

A = Ryx * Rx = [0 1] * [1 0] = [0 -1] [1 0] [0 -1] [1 0]

So, the matrix A = (aij) is:

A = [0 -1] [1 0]

Finally, to find a12, we look at the element in the first row and second column of matrix A. In this case, a12 = -1.