Do not answer if you cannot answer all questions Please answ

Do not answer if you cannot answer all questions. Please answer ALL the questions.

Determine whether the matrix is orthogonal. P = [1/3 2/3 2/3 2/3 1/3 -2/3 2/3 -2/3 1/3] Find PP^T. Is the matrix P is orthogonal? P is orthogonal. P is not orthogonal. Let p_1 = [1 3 2/3 2 3], p_2 = [2 3 1/3 -2 3], and p_3 = [2 3 -2/3 1 3]. If the matrix P is orthogonal, show that the column vectors of the matrix form an orthonormal set. (If the matrix is not orthogonal, enter NOT ORTHOGONAL.) Find p_1 middot p_2. p_1 middot p_2 = Find p_1 middot p_3. p_1 middot p_3 = Find p_2 middot p_3. p_2 middot p_3 = Find ||p_1||. ||p_1|| = Find ||p_2||. ||p_2|| = Find ||p_3||. ||p_3|| = Determine whether the column vectors of the matrix form an orthonormal set. {p_1, p_2, p_3} forms an orthonormal set. {p_1, p_2, p_3} does not form an orthonormal set. P is not an orthogonal matrix.

Solution

A n×n matrix A is an orthogonal matrix if

AA^(T)=I,

where A^(T) is the transpose of A and I is the identity matrix. In particular, an orthogonal matrix is always invertible, and

A^(-1)=A^(T).

Here A^(T)=  

and A^-1=

We can calculate A^-1 using following operations

1. In A multiply the 1st row by 3  

2.add -2/3 times the 1st row to the 2nd row  

3. add -2/3 times the 1st row to the 3rd row

4. multiply the 2nd row by -1

5.add 2 times the 2nd row to the 3rd row

6.multiply the 3rd row by 1/3

7. add -2 times the 3rd row to the 2nd row

8.add -2 times the 3rd row to the 1st row

9.add -2 times the 2nd row to the 1st row

Since A^T = A^-1

The given matrix is orthogonal

PP^T= I

P₁ x P₂ = not possible 3x1 cannot be multiplied with 3x1 It could be multipulied 1x3 or 3x3 with 3x1 etc.

Similarly P₂ x P₃ and P₁ x P₃ could not be calculated.

To calculate a determinant we need the matrix (to be square).

Hence detP₁,detP_2,detP₃ could not be calculated.

p₁,p_2,p₃ does not form an orthonormal set because neither of the 3 matrices are square matrix which is needed in order to check whether matrix is orthonormal or not.

1/3	2/3	2/3
2/3	1/3	-2/3
2/3	-2/3	1/3