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Let A, B, and C be n × n matrices. Which of the following statements is true?

1. If AB = AC, then B = C.
2. If A³ + 2A² + 3A + 5I = 0, then A is invertible.
3. If A² = 0, then A = 0.
4. None of these.

Correct answer: If A³ + 2A² + 3A + 5I = 0, then A is invertible.

Solution

Rewrite as A(A²+2A+3I) = -5I, so A·[-(1/5)(A²+2A+3I)] = I, giving an explicit inverse; hence A is invertible. (AB=AC need not give B=C, and A²=0 need not give A=0, e.g. a nonzero nilpotent matrix.)

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