Given the following matrix.

Question

Given the following matrix A:

1 0 0

a 1 0

b a 1

Which of the following statements must be true for all a, b ∈ R?

a) Ax = 0

b) A^T = A

c) det(A) < 0

d) det(A) = a² - b

Answer 1

Hi Sana,

A good hunch would be "d." The first choice doesn't seem reasonable: multiplying a generic 3x1 vector by A will produce non-zero results in every element.

For the second choice, the transpose A^T is:

1 a b

0 1 a ≠ A

0 0 0

How the determinant could be negative for ALL values of a and b (as answer c states) is hard to imagine, so by POE, answer d makes sense, but we'll calculate it to be sure.

However, det A = 1-a

This is not equivalent to a²- b, and it is not always <0, so the answer is "none of the above."