Question

3. We define the trace of an n × n matrix B = (bij) by the formula tr(B) = Σ bix- a) Is it possible for a 3 × 3 invertible matrix to have trace 0? If so, give an exanple. If not, briefly explain why such a thing is impossible. (b) Give an example of a noninvertible 3 x 3 matrix with all distinct non-zero entries and trace 0.

Answer 1

(a) Yes, it is possible for a 3 x 3 invertible matrix to have trace 0. For a matrix to be invertible, its determinant needs to be non-zero. So, we can build a matrix with trace zero and non-zero determinant. e.g.

and det(A) = 2 i.e. non-zero.

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(b) Example -->   and here det(A) = 0 and hence it is non-invertible.

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