The matrix $A$ is:
$$
\begin{bmatrix}a&b\\c&d\end{bmatrix}
$$
Just started an intro to linear algebra
a class, and I understand what rank means, but I'm not sure how to work with letters.
I do know how to manipulate the matrix into reduced row-echelon form, and can get:
\begin{bmatrix}1&0\\0&1\end{bmatrix}
I do realize this is of rank $2$. Is simply doing $\operatorname{rref}(A)$ enough to solve this? I'm just a little unsure because I know I can say that $A$ can have rank $2$, but must it?
Best Answer
Hint: If $a$ is non-zero, then the first step of row reduction yields $$ \pmatrix{ a&b\\c&d } \to \pmatrix{a &b\\0& d - \frac {bc}a} $$ We will end up with two pivots iff $d - \frac {bc}a \neq 0$.
If $a$ is zero, then show that we end up with two pivots if (and only if) both $b$ and $c$ are non-zero.