Let the matrix
𝑋
be
𝑋
=
[
−
9

4

−
7

𝑘
]
.

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

Question

Let the matrix 
𝑋
 be
𝑋
=
[
−
9
	
4

−
7
	
𝑘
]
.

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

BlackTom AI · Accepted Answer

To determine when X has 0 as an eigenvalue, we need to find when 0 is an eigenvalue of a 2x2 matrix, which occurs precisely when the determinant of X is zero.

Option analysis:
- Option "28
9" (interpreted as the fraction 28/9): This corresponds to solving det(X) = 0. For X = [ [-9, 4], [-7, k] ], the determinant is (-9)*k - (4)*(-7) = -9k + 28. Setting det(X) = 0 gives -9k + 28 = 0, so k = 28/9. Therefore, this option is correct because it matches the required value that makes the determinant zero, ensuring 0 is an eigenvalue.

- There are no additional options provided to evaluate. If another value were present, we would similarly substitute into det(X) = 0 and check whether the resulting k satisfies the equation. In this case, the unique solution to -9k + 28 = 0 is k = 28/9, so any alternative option would be incorrect.

Summary: The determinant condition leads to k = 28/9, so the given option "28
9" is the correct choice. The key idea is that a zero eigenvalue occurs exactly when the determinant is zero for a square matrix, which translates to solving -9k + 28 = 0 for k.

Let the matrix 𝑋 be 𝑋 = [ − 9 4 − 7 𝑘 ] . For 𝑋 to have 0 as an eigenvalue, what must 𝑘 be?单项选择题

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