Theorem 3 ( invertible matrices) Let A be an nxn matrix. The following statements are equivalent. 1. A is invertible 2. The reduced row echelon form of A is 1, 3. The system Ax = b has a unique solution for any nx1 vector b ( * = A" b)
Theorem 3 ( invertible matrices) Let A be an nxn matrix. The following statements are equivalent. 1. A is invertible 2. The reduced row echelon form of A is 1, 3. The system Ax = b has a unique solution for any nx1 vector b ( * = A" b)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Can someone explain these rules? Please answer in best detail possible please.
![**Theorem 3 (Invertible Matrices)**
Let \( A \) be an \( n \times n \) matrix. The following statements are equivalent:
1. \( A \) is invertible.
2. The reduced row echelon form of \( A \) is \( I_n \).
3. The system \( A \vec{x} = \vec{b} \) has a unique solution for any \( n \times 1 \) vector \( \vec{b} \) \(( \vec{x} = A^{-1} \vec{b} )\).
4. The system \( A \vec{x} = \vec{0} \) has only the trivial solution \(( \vec{x} = \vec{0} )\).
5. The columns of \( A \) span \( \mathbb{R}^n \).
6. The columns of \( A \) are independent.
7. The columns of \( A \) form a basis for \( \mathbb{R}^n \).
8. The dimension of the null space of \( A \) is zero.
9. \( \det A \neq 0 \).
10. The matrix \( A \) does not have a zero eigenvalue.
**Homework:**
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Transcribed Image Text:**Theorem 3 (Invertible Matrices)**
Let \( A \) be an \( n \times n \) matrix. The following statements are equivalent:
1. \( A \) is invertible.
2. The reduced row echelon form of \( A \) is \( I_n \).
3. The system \( A \vec{x} = \vec{b} \) has a unique solution for any \( n \times 1 \) vector \( \vec{b} \) \(( \vec{x} = A^{-1} \vec{b} )\).
4. The system \( A \vec{x} = \vec{0} \) has only the trivial solution \(( \vec{x} = \vec{0} )\).
5. The columns of \( A \) span \( \mathbb{R}^n \).
6. The columns of \( A \) are independent.
7. The columns of \( A \) form a basis for \( \mathbb{R}^n \).
8. The dimension of the null space of \( A \) is zero.
9. \( \det A \neq 0 \).
10. The matrix \( A \) does not have a zero eigenvalue.
**Homework:**
[No additional information is provided beneath this heading in the image.]
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