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If A is a 4 x 4 matrix and A is invertible, which of the following statements is not true? O The columns of A form a basis of R4. O The determinant of A # 0. O Zero is an Eigenvalue of A. The rank of A is 4.

If A is a 4 x 4 matrix and A is invertible, which of the following statements is not true? O The columns of A form a basis of R4. O The determinant of A # 0. O Zero is an Eigenvalue of A. The rank of A is 4.

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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Linear Algebra - 4x4 Invertible Matrix Question

### Linear Algebra Quiz: Invertible Matrices **Question:** If \(A\) is a 4 x 4 matrix and \(A\) is invertible, which of the following statements is *not true*? **Options:** 1. \(\bigcirc\) The columns of \(A\) form a basis of \( \mathbb{R}^4 \). 2. \(\bigcirc\) The determinant of \(A \neq 0 \). 3. \(\bigcirc\) Zero is an eigenvalue of \(A\). 4. \(\bigcirc\) The rank of \(A\) is 4. **Explanation:** - An invertible matrix \(A\) has several properties, including: 1. **Basis of \( \mathbb{R}^4 \)**: The columns of an invertible \(4 \times 4\) matrix are linearly independent and span \( \mathbb{R}^4 \). This means they form a basis of \( \mathbb{R}^4 \). 2. **Determinant**: The determinant of an invertible matrix is non-zero. Thus, for matrix \(A\), the determinant should not be zero. 3. **Eigenvalues**: An invertible matrix cannot have zero as an eigenvalue. If zero were an eigenvalue, the matrix would not be invertible. 4. **Rank**: The rank of an invertible \(4 \times 4\) matrix is equal to 4, which means it is full rank. Considering these points, the correct answer to the question "Which of the following statements is not true?" is: \(\bigcirc\) Zero is an eigenvalue of \(A\).
Transcribed Image Text:### Linear Algebra Quiz: Invertible Matrices **Question:** If \(A\) is a 4 x 4 matrix and \(A\) is invertible, which of the following statements is *not true*? **Options:** 1. \(\bigcirc\) The columns of \(A\) form a basis of \( \mathbb{R}^4 \). 2. \(\bigcirc\) The determinant of \(A \neq 0 \). 3. \(\bigcirc\) Zero is an eigenvalue of \(A\). 4. \(\bigcirc\) The rank of \(A\) is 4. **Explanation:** - An invertible matrix \(A\) has several properties, including: 1. **Basis of \( \mathbb{R}^4 \)**: The columns of an invertible \(4 \times 4\) matrix are linearly independent and span \( \mathbb{R}^4 \). This means they form a basis of \( \mathbb{R}^4 \). 2. **Determinant**: The determinant of an invertible matrix is non-zero. Thus, for matrix \(A\), the determinant should not be zero. 3. **Eigenvalues**: An invertible matrix cannot have zero as an eigenvalue. If zero were an eigenvalue, the matrix would not be invertible. 4. **Rank**: The rank of an invertible \(4 \times 4\) matrix is equal to 4, which means it is full rank. Considering these points, the correct answer to the question "Which of the following statements is not true?" is: \(\bigcirc\) Zero is an eigenvalue of \(A\).
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