I have come across this statement in my textbook that the rows and columns of an invertible square matrix are linearly independent, but I am still unsure why.
[Math] Why are the rows and columns of an invertible square matrix linearly independent
linear algebramatrices
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Best Answer
The statement (for columns) is the same as "if $M$ is invertible, then its columns are independent."
Consider the contrapositive statement: "if $M$'s columns are dependent, then $M$ is noninvertible."
If $M$'s columns are dependent, then for some collection of scalars $c_i$, not all equal to $0$, $$c_1\vec{\text{col}_1}+c_2\vec{\text{col}_2}+\cdots+c_n\vec{\text{col}_n}=0$$ which is the same as $$\left[\vec{\text{col}_1}\ \vec{\text{col}_2}\ \cdots\ \vec{\text{col}_n}\right]\begin{bmatrix}c_1\\c_2\\\vdots\\ c_n\end{bmatrix}$$ which demonstrates a nonzero vector that $M$ annihilates. So you can conclude that $M$ is noninvertible (because it maps at least two things to the zero vector.)