Let 𝑋 be a square matrix. Consider the following two statements on 𝑋. I. 𝑋…
2019
Let 𝑋 be a square matrix. Consider the following two statements on 𝑋.
I. 𝑋 is invertible.
II. Determinant of 𝑋 is non-zero.
Which one of the following is TRUE?
Answer: D. I and II are equivalent statements. — Answer: I and II are equivalent statements. Proof: If X is invertible then det(X) ≠ 0. If X has an inverse X^{-1}, then det(X)·det(X^{-1}) = det(I) = 1, so…
- A.
I implies II; II does not imply I.
- B.
II implies I; I does not imply II.
- C.
I does not imply II; II does not imply I.
- D.
I and II are equivalent statements.
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Correct answer: D
Answer: I and II are equivalent statements.
Proof:
If X is invertible then det(X) ≠ 0. If X has an inverse X^{-1}, then det(X)·det(X^{-1}) = det(I) = 1, so det(X) ≠ 0.
If det(X) ≠ 0 then X is invertible. Use the adjugate identity X·adj(X) = det(X)·I. When det(X) ≠ 0, multiplying both sides by 1/det(X) shows that (1/det(X))·adj(X) is a two-sided inverse of X, so X is invertible.
Note: Equivalently, det(X) = 0 exactly when X is singular (not full rank), and det(X) ≠ 0 exactly when X has full rank and is invertible.
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