Linked Questions

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 =...

Let $A$ be a squared matrix, and suppose there exists an $n\in \Bbb N$ in a way that $A^n=0$. Show that $I-A$ is invertible and that $(I-A)^{-1}=I+A+\cdots+A^{n-1}$ I don't have a clue where to start ...

If $A$ is a nilpotent matrix, then how to show that the matrix $I+A$ is invertible.

If $B$ is nilpotent and $B^{k} = 0$ (and B is square), how should I go around proving that $I + B $ is invertible? I tried searching for a formula - $I = (I + B^{k}) = (I + B)(???)$ But I didn't get ...

Suppose that $N \in M_{n,n}(\mathbb{C})$ is nilpotent (that is, $N^k = 0$ for some integer $k > 0$). Show that $I+N$ is invertible, and find its inverse as a polynomial in $N$. I think I got the ...

Suppose that A is a $50 × 50$ matrix such that $A^3 = 0$. Show that the inverse of $I_{50} − A$ is $A^2 + A + I_{50}$. I'm not even sure how to approach this problem. Any help on how to go about ...

Can somebody give me a hint for showing that: The matrix $A+I$ is invertible if there is an integer $k\gt 0$ so that $A^k=0$.

Let $A \in M_{nxn}(\mathbb{C})$, and assume $A^5=\mathbb{O}_{nxn}$ (A zero matrix). Prove that $(I_n-A)^{-1}$ exists and what does it equal to? So the property that $A^5=\mathbb{O}$ is useful here, ...

If matrix $A^3 = O$, then $A - I$ is nonsingular? True or false? I tried to solve it like this: Given a linear transformation $T: V \rightarrow V$, such that $$m(T) = A, m(O) = O, m(I) = I, m(T^3) =...

Given a matrix in $M_{n \times n}$ which is nilpotent. Prove that the Linear Transformation $T:M_{n \times n} \rightarrow M_{n \times n}$ given by $T(X) = X -NX$ is an isomorphism. Shouldn't be ...

The following question was asked on a high school test, where the students were given a few minutes per question, at most: Given that, $$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$ and, $$Q(x)=x^4+x^...

I am preparing for a computer 3D graphics test and have a sample question which I am unable to solve. The question is as follows: For the following 3D transfromation matrix M, find its inverse. Note ...

So my friend and I are working on this and here is what we have so far. We want to show that $\exists \, B$ s.t. $(I+A)B = I$. We considered the fact that $I - A^k = I$ for some positive $k$. Now, if ...

How do I solve this problem? $A$ is $n\times n$ and $A^n=0$. Prove that $I_n-A$ is invertible.

How to find the inverse of $3\times 3$ block upper triangular matrix $$X = \begin{bmatrix} \mathbb{1} & \mathbb{B} & 0\\ 0 & \mathbb{1} & \mathbb{B}\\ 0 & 0 & \mathbb{1} \end{...