Proving the Orthogonality of a Matrix

Math Problem Statement

prove that me matrix is orthogonal

prove that the matrix is orthogonal

To prove that a matrix is orthogonal, we need to show that the matrix $P$ satisfies the condition:

$P^T P = I$

Where:

Let's begin by computing the transpose $P^T$ of the given matrix $P$ :

0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$$ The transpose $$P^T$$ is obtained by swapping the rows and columns of $$P$$: $$P^T = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$$ Next, we compute the product $$P^T P$$: $$P^T P = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$$ Multiplying the two matrices, we get: $$P^T P = \begin{pmatrix} 0 \cdot 0 + 0 \cdot 0 + 1 \cdot 1 & 0 \cdot 1 + 0 \cdot 0 + 1 \cdot 0 & 0 \cdot 0 + 0 \cdot 1 + 1 \cdot 0 \\ 1 \cdot 0 + 0 \cdot 0 + 0 \cdot 1 & 1 \cdot 1 + 0 \cdot 0 + 0 \cdot 0 & 1 \cdot 0 + 0 \cdot 1 + 0 \cdot 0 \\ 0 \cdot 0 + 1 \cdot 0 + 0 \cdot 1 & 0 \cdot 1 + 1 \cdot 0 + 0 \cdot 0 & 0 \cdot 0 + 1 \cdot 1 + 0 \cdot 0 \end{pmatrix}$$ Simplifying each element, we get: $$P^T P = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = I$$ Since $$P^T P = I$$, the matrix $$P$$ is orthogonal. Would you like more details or further explanation on this? Here are 5 related questions to expand your understanding: 1. What are the properties of orthogonal matrices? 2. How do orthogonal matrices relate to rotations and reflections in geometry? 3. Can a matrix with non-real entries be orthogonal? 4. How is the determinant of an orthogonal matrix related to its orthogonality? 5. What happens to the length of a vector when multiplied by an orthogonal matrix? **Tip:** The determinant of an orthogonal matrix is always $$\pm 1$$.

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Matrix Algebra
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P^T P = I

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