Let the matrix X be
X=[4 −1
1 k].

For X to have 0 as an eigenvalue, what must k be?

Question

Let the matrix X be
X=[4	−1
1	k].

For X to have 0 as an eigenvalue, what must k be?

BlackTom AI · Accepted Answer

The problem asks for the value of k that makes 0 an eigenvalue of the matrix X = [ [4, -1], [1, k] ].
First, recall that a scalar λ is an eigenvalue of a square matrix if and only if det(X − λI) = 0. When λ = 0, this condition reduces to det(X) = 0, so we only need the determinant of X to vanish.
Next, compute the determinant of X: det(X) = 4·k − (−1)·1 = 4k + 1. This comes from the standard formula det([a b; c d]) = ad − bc, with a = 4, b = −1, c = 1, d = k.
Set the determinant equal to zero to find k: 4k + 1 = 0 ⇒ 4k = −1 ⇒ k = −1/4.
Therefore, the matrix will have 0 as an eigenvalue precisely when k equals −1/4.
Note on the provided options: there are no explicit answer choices listed in the data. The key conclusion is derived directly from the determinant condition, independent of any distractors; the calculation shows k = −1/4 as the necessary value for zero to be an eigenvalue.

MATH1061/1002/1021 (ND) MATH1061 Canvas Quiz 9

Let the matrix X be X=[4 −1 1 k]. For X to have 0 as an eigenvalue, what must k be?

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