Find the inverse of each of the following orthogonal matrices.
A= [1 0 0
0 cos(theta) sin(theta)
0 -sin(theta) cos(theta)]
sorry its a little displaced but that should be a 3X3 matrix
oh its just the transpose . nevermind
To find the inverse of the given orthogonal matrix A, we need to perform some steps. First, let's denote the given matrix as A:
A = [1 0 0
0 cos(theta) sin(theta)
0 -sin(theta) cos(theta)]
To find the inverse, we'll follow these steps:
Step 1: Transpose the matrix A
Step 2: Replace each entry of the transposed matrix with its negative value
Step 3: Divide each entry of the resulting matrix by the determinant of the original matrix A
Let's go through these steps one by one:
Step 1: Transpose the matrix A
The transpose of a matrix is obtained by interchanging its rows with columns. In this case, the transposed version of matrix A is:
A^T = [1 0 0
0 cos(theta) -sin(theta)
0 sin(theta) cos(theta)]
Step 2: Replace each entry of the transposed matrix with its negative value
To find the negative value of each entry in the transposed matrix A^T, we simply multiply each entry by -1:
-A^T = [-1 0 0
0 -cos(theta) sin(theta)
0 -sin(theta) -cos(theta)]
Step 3: Divide each entry of the resulting matrix by the determinant of the original matrix A
The determinant of matrix A is given by det(A) = 1. So, we divide each entry of the negative transposed matrix (-A^T) by the determinant:
(-A^T)/det(A) = [-1 0 0
0 -cos(theta) sin(theta)
0 -sin(theta) -cos(theta)]
This resulting matrix is the inverse of the original orthogonal matrix A.
Thus, the inverse of the given orthogonal matrix A is:
A^(-1) = [-1 0 0
0 -cos(theta) sin(theta)
0 -sin(theta) -cos(theta)]