Linear Algebra determinant and rank relation

Question

True or False?

If the determinant of a $4 \times 4$ matrix $A$ is $4$ then its rank must be $4$.

Is it false or true?

My guess is true, because the matrix $A$ is invertible.

But there is any counter-example?

Please help me.

Answer 1

You're absolutely correct. The point of mathematical proof is that you don't need to go looking for counterexamples once you've found the proof. Beforehand that's very reasonable, but once you're done you're done.

Determinant 4 is nonzero $\implies$ invertible $\implies$ full rank.

Each of these is a standard proposition in linear algebra.

Answer 2

If it is invertible, it is full-rank. The rank is the dimension of $\operatorname{Im} A$.