$A$ is any square matrix. Which of the following statements are true?
I. $A + A'$ is symmetric.
II. $A - A'$ is skew-symmetric.
III. $AA'$ is symmetric.
IV. $A - A'$ is symmetric.

I and II onlyCorrect answerYour answerI, II and IV onlyCorrect answerYour answerI, II and III onlyCorrect answerYour answerII, III and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$(A + A')' = A' + A$ , $(A - A')' = A' - A = -(A - A')$ and $(AA')' = (A')'A' = AA'$ : I, II and III are true.
A matrix that is skew-symmetric is symmetric only when it is $O$ , so IV fails for most $A$ .

Concept: Symmetric and skew-symmetric combinations
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The rule this testsSymmetric and skew-symmetric matrices

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Ten questions from Matrices, at your own pace.

AAA is any square matrix. Which of the following statements are true?I. A+A′A + A'A+A′ is symmetric.II. A−A′A - A'A−A′ is skew-symmetric.III. AA′AA'AA′ is symmetric.IV. A−A′A - A'A−A′ is symmetric.

Explanation

$A$ is any square matrix. Which of the following statements are true?
I. $A + A'$ is symmetric.
II. $A - A'$ is skew-symmetric.
III. $AA'$ is symmetric.
IV. $A - A'$ is symmetric.