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Suppose $A$ is an $(n \times n)$ matrix over $\mathbb{R}$ such that $A^{p}=0$ for some positive integer $p$.

1. Prove that $I+A$ is an invertible matrix, where $I$ is the $(n \times n)$ identity matrix.
2. Find the characteristic polynomial of $A$.

First I am showing that one proof which will help to prove $I+A$ is an invertible matrix.

Statement :- Show that if $\lambda$ is an eigen value of A , then $(1+\lambda)$ is an eigen value of $I+A$ .

Proof :-  Say $x$ is an eigen vector of A corresponding to eigen value $\lambda$ ,

$(I+A)x=Ix+Ax=1x+\lambda x=(1+\lambda)x$

So,$x$ is an eigen vector of $(A+I)$ corresponding to eigen value $(1+\lambda)$.

Now given matrix A has the property $A^{p}=0$ for some positive integer $p$. We call this matrix nilpotent matrix as well .

Statement :- If A is nilpotent matrix ,then All its eigen values are zero .

Proof:- Given $A^{p}=0$ . Let $\lambda$ is an eigen value of $A$ with eigen vector $x$.

So , $Ax=\lambda x$

or,   $A^{p}x=A^{p-1}Ax=A^{p-1}\lambda x=\lambda A^{p-1}x=\lambda A^{p-2}Ax=\lambda A^{p-2}\lambda x=\lambda^{2} A^{p-2}x$$=……….=\lambda^{p}x$

So, $A^{p}x=\lambda^{p}x$

Since $A^{p}=0$ we can conclude either $\lambda^{p}=0$ or $x=0$ now as $x$ is eigen vector and eigen vector cannot be zero so ,

$\lambda^{p}=0$ so, $\lambda=0$ .

Now A has all $n$ eigen value as $0,0,0,0,0,…0$ .So A+I has all eigen value as n 1’s $1,1,1,1,1,...1.$

Now we know determinant of a matrix is product of all its eigen values .So, determinant of $\left | A+I \right |=1*1*1*1*1*.......*1=1$

Now We know that a matrix is  invertible if its determinant is non-zero , so $A+I$ is inverible .

Part 2:-  We know that ,the roots of the characteristic polynomials are eigen values of that matrix . If the n eigen values of the matrix is $\lambda _{1},\lambda _{2},\lambda _{3},...........,\lambda _{n}$ then characteristic polynomial is ,

$f(x)=(x-\lambda _{1})(x-\lambda _{2})(x-\lambda _{3}).......(x-\lambda _{n})$

Now Given Matrix $A$ is nilpotent matrix whose eigen values are all zero (already proved) .

So characteristic polynomial of A is ,

$f(x)=(x-0)(x-0)(x-0)........(x-0)=x^{n}$

$\large f(x)=x^{n}$

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