Math Problem Statement

Suppose {v1, v2, v3} is an orthogonal set. Let Q = (v1, v2, v3) be the matrix whose columns are the given vectors. Which of the following statements is true of Q^T Q?

Solution

Let's analyze the given problem step by step:

We are given an orthogonal set of vectors {v1,v2,v3}\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}, and a matrix Q\mathbf{Q} formed by these vectors as columns. The problem asks us to determine which statement about QTQ\mathbf{Q}^T \mathbf{Q} is true.

Key Observations:

  1. 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.
  2. Matrix QTQ\mathbf{Q}^T \mathbf{Q}: This is the Gram matrix for the vectors. For orthonormal vectors, QTQ\mathbf{Q}^T \mathbf{Q} would be the identity matrix, but for merely orthogonal vectors, it might not be.

Checking the Options:

  1. 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 QTQ\mathbf{Q}^T \mathbf{Q} could be zero.

  2. It is a symmetric matrix: This is true. For any matrix Q\mathbf{Q}, the matrix QTQ\mathbf{Q}^T \mathbf{Q} is always symmetric, since (QTQ)T=QTQ(\mathbf{Q}^T \mathbf{Q})^T = \mathbf{Q}^T \mathbf{Q}.

  3. It is the identity matrix: This is false. QTQ\mathbf{Q}^T \mathbf{Q} 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.

  4. It is invertible: This is false. If one of the vectors is the zero vector, then QTQ\mathbf{Q}^T \mathbf{Q} 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:

  1. What is the difference between orthogonal and orthonormal vectors?
  2. How can you construct an orthonormal set from an orthogonal set?
  3. What are the properties of a matrix formed by orthogonal vectors?
  4. When is a matrix QTQ\mathbf{Q}^T \mathbf{Q} invertible?
  5. How does the determinant of a matrix relate to its invertibility?

Tip: Remember, the Gram matrix QTQ\mathbf{Q}^T \mathbf{Q} 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)