Given a matrix $A$ of rank $n$, show that $\det(\operatorname{adj}(A))=\det(A)^{n-1}$ and $\operatorname{adj}(\operatorname{adj}(A))=(\det A)^{n-2}A$
131212
Prove that if $A$ is regular then $\operatorname{adj}(\operatorname{adj}(A)) = (\det A)^{n-2} A$
65395
Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
20331
Determinant of a $2\times 2$ block matrix
26570
Prove $\operatorname{rank}(AB) = \operatorname{rank}(B)$
57771
Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.
78224
Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$
19203
Is the rank of a matrix the same of its transpose? If yes, how can I prove it?
7376
Find $\det(A^{2}+A^{T})$ when eigenvalues are $1,2,3$
74341
Proving an invertible matrix which is its own inverse has determinant $1$ or $-1$
119435
Rank of the difference of matrices
30866
Relation between rank and number of distinct eigenvalues of a matrix
8554
Proof: $\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A$ for $A \in \mathbb{R}^{n\times n}$
1865
Sub-determinants of an orthogonal matrix
21758
Rank of the $n \times n$ matrix with ones on the main diagonal and $a$ off the main diagonal
87425
Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries.
6905
Rank of product of a matrix and its transpose
149086
Determining $\mbox{tr}(A)$ and $\det(A)$ as functions of the eigenvalues of a matrix