Math Problem Statement
Solution
Let's analyze the given problem step by step:
We are given an orthogonal set of vectors , and a matrix formed by these vectors as columns. The problem asks us to determine which statement about is true.
Key Observations:
- Orthogonal Set: This implies that the vectors are orthogonal to each other, but they are not necessarily normalized. Hence, the vectors may not form an orthonormal set.
- Matrix : This is the Gram matrix for the vectors. For orthonormal vectors, would be the identity matrix, but for merely orthogonal vectors, it might not be.
Checking the Options:
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The determinant must be positive: This is false. If one of the vectors is the zero vector (allowed for an orthogonal set), the determinant of could be zero.
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It is a symmetric matrix: This is true. For any matrix , the matrix is always symmetric, since .
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It is the identity matrix: This is false. would be the identity matrix only if the vectors are orthonormal, not just orthogonal. Since orthogonal vectors need not be normalized, this condition doesn’t hold.
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It is invertible: This is false. If one of the vectors is the zero vector, then is singular (non-invertible), since its determinant would be zero.
Conclusion:
The correct statement is "It is a symmetric matrix."
Would you like more details on orthogonal matrices or anything else? Here are five related questions you might find interesting:
- What is the difference between orthogonal and orthonormal vectors?
- How can you construct an orthonormal set from an orthogonal set?
- What are the properties of a matrix formed by orthogonal vectors?
- When is a matrix invertible?
- How does the determinant of a matrix relate to its invertibility?
Tip: Remember, the Gram matrix gives important information about the lengths and angles between vectors!
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Orthogonality
Determinants
Formulas
Q^T Q = Gram matrix of the vectors
Theorems
Symmetry of Q^T Q
Conditions for matrix invertibility
Determinants and singularity
Suitable Grade Level
Undergraduate (Linear Algebra course)
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