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Which of these statements are true?

(I) If $A$ is a matrix of full rank, then $A$ is invertible.

(II) If $A$ can be expressed as a product of elementary matrices, then $A$ is of full rank.

(III) If $A$ is symmetric, then a basis for the row space of $A$ is also a basis for the column space of $A$.

Understanding that equivalent statements are only for $A_{n \times n}$ matrix, so (I) cannot be true. I have put (II) and (III) to be the correct statements, am I correct?

Vee Hua Zhi
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Xon
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1 Answers1

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(I) is, in fact, an "if and only if", so it's more than only true. You can find a proof here: Proof - Square Matrix has maximal rank if and only if it is invertible

(II) is also true. Note that $\text{det}(A\cdot B) = \text{det}A \cdot \text{det} B$. The determinant of a elementary matrix is never zero because they are invertible, so the product of determinants of the elementary matrices is not zero, implying that $\text{det}A \neq 0$, therefor invertible.

(III) holds trivially, but the reciprocal is not true.

As the statements are talking about invertibility and base span of a matrix you can assume that $A \in M_{n}$ (is a square matrix).

Joan Salvà
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  • Hi, thanks for the explanation. Do you mind explaining to me why III is false? – Xon Oct 21 '18 at 09:48
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    (III) is not false. I wrote "it holds trivially", which means it can be prooved very easily. I just said that the reciprocal is not true, which means that if the row space and the column space have the same base, then the matrix $A$ doesn't have to be symmetric. – Joan Salvà Oct 21 '18 at 11:50
  • I see, sorry for the misunderstanding as English is not my 1st language. Thank you for the explanation! – Xon Oct 21 '18 at 11:55
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    Don't worry! Hope I could help – Joan Salvà Oct 21 '18 at 12:32