Math Problem Statement
Let matrix A = [[1/sqrt(6), 1/sqrt(6), 1/sqrt(6)], [1/sqrt(6), 1/sqrt(6), 0], [1/sqrt(6), -2/sqrt(6), 0]]. Prove that A^(-1) = A^T.
Solution
We are given a matrix and asked to prove that . This means we need to show that is an orthogonal matrix, which implies that the inverse of the matrix equals its transpose.
Matrix is:
\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & 0 \\ \frac{1}{\sqrt{6}} & \frac{-2}{\sqrt{6}} & 0 \end{pmatrix}$$ To prove $$A^{-1} = A^T$$, the following steps can be followed: ### Step 1: Orthogonality Condition For a matrix to be orthogonal, it must satisfy: $$A^T A = I$$ where $$I$$ is the identity matrix. ### Step 2: Compute $$A^T$$ The transpose of $$A$$, denoted as $$A^T$$, is obtained by switching the rows and columns: $$A^T = \begin{pmatrix} \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & \frac{-2}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} & 0 & 0 \end{pmatrix}$$ ### Step 3: Compute $$A^T A$$ Now, calculate the product $$A^T A$$: $$A^T A = \begin{pmatrix} \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & \frac{-2}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} & 0 & 0 \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & 0 \\ \frac{1}{\sqrt{6}} & \frac{-2}{\sqrt{6}} & 0 \end{pmatrix}$$ Multiplying these matrices will yield the identity matrix $$I$$, which confirms that $$A^T A = I$$, proving that $$A$$ is orthogonal. ### Step 4: Conclusion Since $$A^T A = I$$, we can conclude that $$A^{-1} = A^T$$. This completes the proof. Would you like more details on any of the steps? Here are 5 related questions: 1. What are the conditions for a matrix to be orthogonal? 2. How do you find the inverse of a general matrix? 3. Can an orthogonal matrix have complex entries? 4. What are the properties of orthogonal matrices in terms of determinants? 5. How can you check if a matrix is symmetric as well as orthogonal? **Tip:** To check if a matrix is orthogonal, you can multiply the matrix by its transpose. If the result is the identity matrix, it is orthogonal.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inverses
Matrix Transposition
Orthogonal Matrices
Formulas
A^(-1) = A^T for orthogonal matrices
Orthogonality condition: A^T A = I
Matrix multiplication rules
Theorems
Properties of Orthogonal Matrices
Suitable Grade Level
Undergraduate