If $A \in M(n , \mathbb R) $ be such that $A^2=A$ , then is it true that $rank (A)=Trace(A)$ ?
15197
Relation between trace and rank for projection matrices
155451
Upper bound for the rank of a nilpotent matrix , if $A^2 \ne 0$
19203
Is the rank of a matrix the same of its transpose? If yes, how can I prove it?
122853
Show that $\operatorname{rank}(A) = \operatorname{rank}(B)$
17430
If $\operatorname{rank}\left( \begin{bmatrix} A &B \\ C &D \end{bmatrix}\right)=n$ prove that $\det(AD)=\det(BC)$
65224
Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?
67058
Prove that $\text{rank}(X^TX)=\text{rank}(X)$
55461
How to prove that if $a$ belongs to $\mathbb R$, such that $0\leq a \leq\epsilon$, then $a = 0$
137093
What is the codimension of matrices of rank $r$ as a manifold?
71040
If $ \operatorname{Tr}(M^k) = \operatorname{Tr}(N^k)\;\forall 1\leq k \leq n$, then how do we show the $M$ and $N$ have the same eigenvalues?
131212
Prove that if $A$ is regular then $\operatorname{adj}(\operatorname{adj}(A)) = (\det A)^{n-2} A$
139685
If $A^2$ and $B^2$ are similar matrices, do $A$ and $B$ have to be similar?
34714
What is the relationship between $\ker(A), \ker(A^2), \ldots, \ker(A^n)$?
39585
Connection between rank and matrix product
102817
$A,B\in M_{n}(\mathbb{R})$ so that $A>0, B>0$, prove that $\det (A+B)>\max (\det(A), \det(B))$
103061
Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?
133756
Geometric interpretation of $\det(A^T) = \det(A)$