Matrices and Determinants Questions and Answers

Answer: C

Omitting the transposition leaves every entry in the wrong position, so the result is the transpose of the true inverse. It coincides with the correct answer only when the matrix is symmetric, which is why the slip sometimes escapes notice.

Answer: D

Correct answer: both the assertion and the reason are true and the reason explains the assertion

Both are true and the reason gives the proof. Interchange the two identical rows: the matrix is unchanged, so its determinant is unchanged, but the interchange rule says the determinant has been negated. Hence \( D = -D \), which forces \( D = 0 \). The argument is a complete derivation, not merely a related fact.

Answer: A

Distinct eigenvalues do guarantee independence and so make the matrix diagonalisable, but they are sufficient rather than necessary. A repeated eigenvalue may still supply enough independent eigenvectors, as the identity matrix shows.

Answer: B

For a two by two matrix the pair of relations determines the roots completely, and for larger matrices they still provide two independent arithmetic checks. Both hold whatever the symmetry or the multiplicity of the eigenvalues.

Answer: C

Similar matrices represent the same map in different bases, so the characteristic polynomial and with it the eigenvalues, trace and determinant all survive. The entries, the triangular form and the basis itself all change.