Suppose $A$ be an $n×n$ real matrix be such that $A^k= O$ for some $k  \in \Bbb N$ then prove that $I+A$ is invertible.  Please do not use eigenvalues.
5478
If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit.   I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) = 1 = (u+a)x$. After tried several elements and manipulated $ua = au$, I still couldn't find any clue. Can anybody give me a hint?
4810
The $2 \times 2$ matrix ${A}$ satisfies ${A}^2 - 4 {A} - 7 {I} = {0}$ where ${I}$ is the $2 \times 2$ identity matrix. Prove that ${A}$ is invertible.  I have tried to solve it like a quadratic, but that doesn't work.  Any help is appreciated!
99031
Let $A$ and $B$ be $n$ $\times$ $n$ real matrices such that $AB$=$BA$=$0$ and $A+B$ is invertible. Which of the following are always true?  $1$. rank $(A)$=rank $(B)$.  $2$. rank $(A)$$+$rank $(B)$ =$n$  $3$. nullity $(A)$ $+$ nullity $(B)$ = $n$  $4$. $A-B$ is invertible.  I am not getting how to do. Atleast it is clear that Rank$(A+B)$= $n$. help me!
68240
Let $A$ be an $m\times n$ matrix. Prove that all eigenvalues of $A^*A$ are non-negative.
88515
Let A be some $2x2$ matrix with real entries. Prove that $A^T$$A$ = $I$ if and only if $A$ is the rotation matrix or the reflection matrix.  My Progress: It can be shown that if $A$ is either the rotation or reflection matrix, then $A^T$$A$ = $I$ holds by matrix multiplication. Where I get suck is showing that if $A$ is a $2x2$ orthogonal matrix, then $A$ must either be the Rotation or Reflection Matrix. I suppose that since $A$ is orthogonal, it is distance preserving - and the only $2x2$ matrices that preserve distance are the Rotation and Reflection Matrices, but this isn't really a proof.  Any advice and help would be much appreciated.
22697
Let $A$ be an invertible square matrix. Prove that $$ |A + uv^T| = (1 + v^T A^{-1} u) |A|. $$   My attempt $|A + uv^T| = \left|A (I + A^{-1}uv^T)\right| = |A| |I + A^{-1} uv^T|$ I'm trying to use the following fact, $$(I + ab^T)^{-1} = I - \frac{1}{1+b^T a}ab^T$$ but it seems quite tricky because of the $A^{-1}$ part.
31703
If $A$ and $B$ are $n\times n$ matrices, and $AB$ is invertible then $A$ and $B$ are invertible.    I started out by writing that since $AB$ is invertible, then for the equation $ABx=b$ has a unique solution for any $b$, $x=(AB)^{-1}b$.  Then I don't know how to break up the composition of AB to prove they are each individually similar.
59974
$A$ is an $n\times n$ matrix with $\|A\|≤a&lt;1 $ . I need to prove that the matrix $(I-A)$ is invertible with $\|(I-A)^{-1}\|\le\frac1{(1-a)}$.   It doesn't say anything more. The norm makes me confuse. How can we start to solve this. Could you please help?
11744
$A$ is a real orthogonal matrix and $(I + A)$ is non-singular. Prove that $ (I+A)^{-1}(I-A)$ is a skew-symmetric matrix.  Attempt:  $[(I+A)^{-1}(I-A)]^t=(I-A)^t((I+A)^{-1})^t=(I-A)^t((I+A)^{t})^{-1}=(I-A^t)   ((I+A^t))^{-1}$   Please help me further.
34125
Establish if the matrix $A\in M_3 (\Bbb{Z}_9)$  $$     \begin{bmatrix}     1 &amp; 2 &amp; 3 \\     2 &amp; 3 &amp; 0 \\     0 &amp; 3 &amp; 6     \end{bmatrix} $$  is invertible, and if so, find the inverse matrix.
11622
Given a ring $R$ (with unity), prove that if $a \in R$ is nilpotent, then $a+1,a-1$ are both invertible.   Suppose $a^n=0$. Then $1=1-a^n=(1-a)(1+a+\ldots+a^{n-1})$, so $1-a$ is invertible.  If $n$ is odd we can use $1+a^n=(1+a)(1-a+\ldots+a^{n-1})$.  But what if $n$ is even?
46403
assume $A,B\in M_n(F)$ if  $I-AB$ be  invertible then  how  to prove $I-BA$ is invertible and how find inverse of $I-BA$   Thanks in advance
22526
Let $A=[a_{ij}]_{n×n}$ be the matrix defined by letting $a_{ij}$ be the rational number such that $$a_{ij}=\gcd(i,j ).$$ How prove that $A$ is invertible, and compute $\det(A)$? thanks in advance
38947
Say if $A$ is an $n \times  n$ matrix, why is it that if $A^{T}A$ is positive definite, the matrix $A$ is then invertible? All I know is $A^{T}A$ gives a symmetric matrix but what does $A^{T}A$ is positive definite tell or imply or hint about the matrix $A$ itself that leads to the fact that it will be invertible?
53901
I saw this tucked away in a  and am asking this question to preserve (and advertise?) it. It's a nice problem!  Problem: Suppose $A$ and $B$ are real $n\times n$ matrices with $A^2+B^2=AB$. If $AB-BA$ is invertible, prove $n$ is a multiple of $3$.
18833
Let $A$ be a complex matrix of size $n$ if for some positive integer $k$ , $A^k=I_n$ , then is $A$ diagonalizable ?
87552
Let $A$ be a diagonalizable matrix and $\lambda$ be an eigenvalue of A. Prove that rank($\lambda I - A$) = rank(($\lambda I - A)^2$).  The hint they provided is that for any invertible matrix B, rank(AB) = rank (A) and rank(BA) = rank(A).