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If a 6 x 3 matrix A has rank 3, find nullity A, rank A, and rank AT.

If a 6 x 3 matrix A has rank 3, find nullity A, rank A, and rank AT.

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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**Problem Statement:** If a \(6 \times 3\) matrix \(A\) has rank 3, find the nullity of \(A\), rank \(A\), and rank \(A^T\). **Solution Explanation:** 1. **Rank \(A\):** The rank of matrix \(A\) is already given as 3. 2. **Nullity of \(A\):** The nullity of a matrix is calculated as the difference between the number of columns and the rank. Since \(A\) is a \(6 \times 3\) matrix, it has 3 columns. Nullity \(A = 3 - \text{rank} A = 3 - 3 = 0\). 3. **Rank \(A^T\):** The rank of a matrix is equal to the rank of its transpose. Therefore, rank \(A^T = \text{rank} A = 3\). This problem demonstrates fundamental concepts in linear algebra related to matrix dimensions and rank-nullity theorem.
Transcribed Image Text:**Problem Statement:** If a \(6 \times 3\) matrix \(A\) has rank 3, find the nullity of \(A\), rank \(A\), and rank \(A^T\). **Solution Explanation:** 1. **Rank \(A\):** The rank of matrix \(A\) is already given as 3. 2. **Nullity of \(A\):** The nullity of a matrix is calculated as the difference between the number of columns and the rank. Since \(A\) is a \(6 \times 3\) matrix, it has 3 columns. Nullity \(A = 3 - \text{rank} A = 3 - 3 = 0\). 3. **Rank \(A^T\):** The rank of a matrix is equal to the rank of its transpose. Therefore, rank \(A^T = \text{rank} A = 3\). This problem demonstrates fundamental concepts in linear algebra related to matrix dimensions and rank-nullity theorem.
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