- #1

JD_PM

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- 158

- Homework Statement:
- Prove or give a counterexamples for the following statements

- Relevant Equations:
- N/A

Hi guys! :)

I was solving some linear algebra true/false (i.e. prove the statement or provide a counterexample) questions and got stuck in the following

a) There is no ##A \in \Bbb R^{3 \times 3}## such that ##A^2 = -\Bbb I_3## (typo corrected)

I think this one is true, as there is no squared matrix that yields a -ive number. Is that enough justification?

b) If the matrix elements of a matrix ##A## are all integers and ##\det(A) = \pm 1## then the matrix elements of ##A^{-1}## are also integers.

This seems to be true as I tried to find a counterexample with several ##2 \times 2## matrices with ##\det(A) = \pm 1## and all their inverses contained integer numbers only. However I do not really see how to actually prove the statement. Could you please provide a hint?

c) If the matrix elements of a square matrix ##A## are all zero or 1 then ##\det A =1, 0## or ##-1##.

As above, I tried to find a counterexample but I found none so I suspect the statement is true. But as with b), I do not see how to actually prove it. Could you please provide a hint for this one as well?

Thanks!

I was solving some linear algebra true/false (i.e. prove the statement or provide a counterexample) questions and got stuck in the following

a) There is no ##A \in \Bbb R^{3 \times 3}## such that ##A^2 = -\Bbb I_3## (typo corrected)

I think this one is true, as there is no squared matrix that yields a -ive number. Is that enough justification?

b) If the matrix elements of a matrix ##A## are all integers and ##\det(A) = \pm 1## then the matrix elements of ##A^{-1}## are also integers.

This seems to be true as I tried to find a counterexample with several ##2 \times 2## matrices with ##\det(A) = \pm 1## and all their inverses contained integer numbers only. However I do not really see how to actually prove the statement. Could you please provide a hint?

c) If the matrix elements of a square matrix ##A## are all zero or 1 then ##\det A =1, 0## or ##-1##.

As above, I tried to find a counterexample but I found none so I suspect the statement is true. But as with b), I do not see how to actually prove it. Could you please provide a hint for this one as well?

Thanks!

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