True or False: Suppose that A is a 3x3 matrix whose eigenvalues are -3, 5 and 1/2. Then it must be the case that A is an invertible matrix.判断题
A
True
B
False

Question

True or False: Suppose that A is a 3x3 matrix whose eigenvalues are -3, 5 and 1/2. Then it must be the case that A is an invertible matrix.

BlackTom AI · Accepted Answer

Consider the statement and its implications by analyzing what eigenvalues tell us about invertibility.
Option 1 (True): If a square matrix has an eigenvalue of 0, then the matrix is singular and not invertible; equivalently, a matrix is invertible precisely when none of its eigenvalues are zero. In this case, the eigenvalues are -3, 5, and 1/2, none of which are zero. Therefore, the determinant (which equals the product of the eigenvalues for a square matrix) is (-3) × 5 × (1/2) = -15/2 ≠ 0, which confirms invertibility.
Option 2 (False): If one were to claim the matrix must not be invertible, the reasoning would require a zero eigenvalue or a zero determinant. Since all given eigenvalues are nonzero, this line of reasoning is flawed; there is no evidence here that would force a zero determinant or a noninvertible transformation, so this option is incorrect.
Additional perspective: Invertibility is equivalent to the absence of zero in the spectrum of the matrix. Here, the spectrum is { -3, 5, 1/2 }, which clearly excludes 0, reinforcing the conclusion that A is invertible.
A potential misconception to watch for: believing that a matrix with nonzero eigenvalues might still fail to be invertible due to some other property. However, for square matrices over the reals or complexes, nonzero eigenvalues guarantee a nonzero determinant and thus invertibility, so that idea does not apply here.
In summary, the presence of only nonzero eigenvalues directly contradicts any claim of noninvertibility and supports the invertibility of A.

True or False: Suppose that A is a 3x3 matrix whose eigenvalues are -3, 5 and 1/2. Then it must be the case that A is an invertible matrix.判断题

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