How to prove $\det (A^{-1}) = \frac{1}{\det A}$ without using the property that $\det (AA^{-1})=\det (A) \det(A^{-1})$?
113518
Determinant of the inverse matrix
15164
Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
98135
Show that the determinant of $A$ is equal to the product of its eigenvalues
83662
Prove that $ a $ divides $1^{a^{n}} + 2^{a^{n}} + ... + (a-1)^{a^{n}}$?
68705
Prove that $A\ge0, B\ge0$ and $A\ge B$ implies $B^{-1}\ge A^{-1}$
14646
Prove $\det(kA)=k^n\det A$
154001
$(g\circ f)^{-1}=f^{-1}\circ g^{-1}$
57771
Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.
91000
Prove that $\frac{1}{a+1}0$
90486
If $(A-B)^2=O_2$ then $\det(A^2 - B^2)=(\det(A) - \det(B))^2$
125348
Does $\det(A + B) = \det(A) + \det(B)$ hold?
85167
When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$
106305
Prove/disprove: if $\det(A+X) = \det(B + X)$ for all $X$, then $A=B$
78224
Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$
32487
Proving $\det \left( \begin{smallmatrix} A & -B \\ B & A \end{smallmatrix} \right) =|\det(A+iB)|^2$
84522
A better proof for $\det(P) = \pm1$ if $P$ is an orthogonal matrix
87425
Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries.