Let $A$ be an nxn matrix such that the first $3$ rows of $A$ are linearly independent and the first $5$ columns of $A$ are linearly independent.

Question

Let $A$ be an $n \times n$ matrix such that the first $3$ rows of $A$ are linearly independent and the first $5$ columns of $A$ are linearly independent. Which of the following statements are true?

1. $A$ has at least $5$ linearly independent rows

2. $3\leq rankA \leq 5 $

3. $rank A \geq 5. $

4. $rank(A^2)\geq 5. $

Using the concept that row rank(A)=column rank(A) I could conclude that option 1 and option 3 are correct. Choose $A=I_n, n>6.$ We can conclude that option 2 is incorrect. I couldn't able to find a counter example for option 4. How to disprove option 4? Please help me.

Answer 1

Suppose $n=10$. Then because the matrix is square, $$\mathrm{rank}~ A^2 \geq 2~\mathrm{rank}~A - (\text{# of columns})$$ Then $$\mathrm{rank}~A^2 ≥ 2(5)-10 = 0$$ So option 4 is wrong.