Question

Let \( A \) be a \( 3 \times 3 \) real matrix and let \( I_3 \) be the \( 3 \times 3 \) identity matrix. Which one of the following statements is NOT true?

• A. If the row-reduced echelon form of \( A \) is \( I_3 \), then zero is not an eigenvalue of \( A \)
• B. If zero is not an eigenvalue of \( A \), then the row-reduced echelon form of \( A \) is \( I_3 \)
• C. If \( A \) has three distinct eigenvalues, then the row-reduced echelon form of \( A \) is \( I_3 \)
• D. If the system of equations \( Ax = b \) has a solution for every \( 3 \times 1 \) real column vector \( b \), then the row-reduced echelon form of \( A \) is \( I_3 \)

Hint:

If a matrix is invertible (no zero eigenvalues), its row-reduced echelon form is \( I_3 \). However, the converse is not true; a matrix can be invertible without having its RREF as \( I_3 \).

Answer 1

The Correct Option is B

Step 1: Understanding the problem.
We are given a \( 3 \times 3 \) matrix \( A \) and the identity matrix \( I_3 \). We need to analyze the truth of four statements related to the row-reduced echelon form (RREF) of \( A \), eigenvalues of \( A \), and solutions to the system \( Ax = b \). Step 2: Analyzing the statements.
(A) If the row-reduced echelon form of \( A \) is \( I_3 \), then zero is not an eigenvalue of \( A \):
This statement is true. If the row-reduced echelon form (RREF) of \( A \) is the identity matrix \( I_3 \), it means that \( A \) is invertible and has full rank. An invertible matrix cannot have zero as an eigenvalue because having a zero eigenvalue would imply that \( A \) is singular and not invertible. Thus, zero cannot be an eigenvalue.
(B) If zero is not an eigenvalue of \( A \), then the row-reduced echelon form of \( A \) is \( I_3 \):
This statement is NOT true. While it is true that if the row-reduced echelon form of \( A \) is \( I_3 \), then \( A \) is invertible and thus does not have zero as an eigenvalue, the converse is not true. A matrix may have no zero eigenvalue (i.e., it is invertible) but still not have the row-reduced echelon form as \( I_3 \). For example, \( A \) could be a matrix that is not in row-reduced form, but it still has no zero eigenvalue.
(C) If \( A \) has three distinct eigenvalues, then the row-reduced echelon form of \( A \) is \( I_3 \):
This statement is true. If \( A \) has three distinct eigenvalues, it implies that \( A \) is diagonalizable and has full rank. Hence, its row-reduced echelon form must be \( I_3 \), the identity matrix, because it is invertible and has no zero eigenvalue.
(D) If the system of equations \( Ax = b \) has a solution for every \( 3 \times 1 \) real column vector \( b \), then the row-reduced echelon form of \( A \) is \( I_3 \): This statement is true. If the system \( Ax = b \) has a solution for every possible \( b \), it means that \( A \) is invertible. In this case, the row-reduced echelon form of \( A \) must be \( I_3 \), because an invertible matrix has full rank and its row-reduced form is the identity matrix.
Step 3: Conclusion.
The false statement is (B), as having no zero eigenvalue does not necessarily imply that the row-reduced echelon form of \( A \) is \( I_3 \).