# 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.

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.

• Think of a nilpotent matrix
– Jan
Commented Feb 15, 2022 at 6:53
• okay. thank you very much. Commented Feb 15, 2022 at 7:53

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.