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Suppose A is a 3xn matrix whose columns span R3. Explain how to construct an nx3 matrix D such that AD = I,: %3D

Suppose A is a 3xn matrix whose columns span R3. Explain how to construct an nx3 matrix D such that AD = I,: %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Matrix Construction in Linear Algebra** **Problem Statement:** Suppose \( A \) is a \( 3 \times n \) matrix whose columns span \(\mathbb{R}^3\). Explain how to construct an \( n \times 3 \) matrix \( D \) such that \( AD = I_3 \). **Explanation:** To achieve this, we need to understand the concept of a right inverse. Since the columns of \( A \) span \(\mathbb{R}^3\), \( A \) has full row rank, meaning it is possible to find a matrix \( D \) such that when \( A \) is multiplied by \( D \), the product is the \( 3 \times 3 \) identity matrix \( I_3 \). **Steps:** 1. **Identify the Full Row Rank:** Since the columns of \( A \) span \(\mathbb{R}^3\), the matrix \( A \) must have full row rank. This means the rank of \( A \) is 3. 2. **Construct the Matrix \( D \):** Use the fact that \( A \) has full row rank to construct a matrix \( D \) that satisfies the equation \( AD = I_3 \). Generally, to find such a \( D \) for matrices where the number of columns exceeds the number of rows, the Moore-Penrose pseudoinverse could be utilized. However, the specific method might depend on the properties of \( A \). 3. **Verification:** Calculate the product \( AD \) and check if the result is the identity matrix \( I_3 \). This step is crucial to ensure that \( D \) has been correctly constructed. By following these steps, a suitable matrix \( D \) can be derived, achieving the desired equality.
Transcribed Image Text:**Matrix Construction in Linear Algebra** **Problem Statement:** Suppose \( A \) is a \( 3 \times n \) matrix whose columns span \(\mathbb{R}^3\). Explain how to construct an \( n \times 3 \) matrix \( D \) such that \( AD = I_3 \). **Explanation:** To achieve this, we need to understand the concept of a right inverse. Since the columns of \( A \) span \(\mathbb{R}^3\), \( A \) has full row rank, meaning it is possible to find a matrix \( D \) such that when \( A \) is multiplied by \( D \), the product is the \( 3 \times 3 \) identity matrix \( I_3 \). **Steps:** 1. **Identify the Full Row Rank:** Since the columns of \( A \) span \(\mathbb{R}^3\), the matrix \( A \) must have full row rank. This means the rank of \( A \) is 3. 2. **Construct the Matrix \( D \):** Use the fact that \( A \) has full row rank to construct a matrix \( D \) that satisfies the equation \( AD = I_3 \). Generally, to find such a \( D \) for matrices where the number of columns exceeds the number of rows, the Moore-Penrose pseudoinverse could be utilized. However, the specific method might depend on the properties of \( A \). 3. **Verification:** Calculate the product \( AD \) and check if the result is the identity matrix \( I_3 \). This step is crucial to ensure that \( D \) has been correctly constructed. By following these steps, a suitable matrix \( D \) can be derived, achieving the desired equality.
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