1. ## induction proof

Let $\displaystyle A \in M_{n}(mathbb{R})$.
If $\displaystyle A$ is invertible, show that $\displaystyle A^{n} \neq 0$ for all positive integer $\displaystyle n$.

I started with $\displaystyle n=1$, then $\displaystyle A^{n}=A^{1} \neq 0$ since $\displaystyle A$ is invertible.

Assume that $\displaystyle A^{n} \neq 0$ for some positive integer $\displaystyle n$.

Now, how to show that $\displaystyle A^{n+1} \neq 0$???
$\displaystyle A^{n+1}=A^{n}A^{1}$ but, even though $\displaystyle A^{n}, A^{1}$ nonzero matrix, $\displaystyle A^{n+1}$ can be zero matrix also.

I'm stuck here.

2. Originally Posted by deniselim17
Let $\displaystyle A \in M_{n}(mathbb{R})$.
If $\displaystyle A$ is invertible, show that $\displaystyle A^{n} \neq 0$ for all positive integer $\displaystyle n$.

I started with $\displaystyle n=1$, then $\displaystyle A^{n}=A^{1} \neq 0$ since $\displaystyle A$ is invertible.

Assume that $\displaystyle A^{n} \neq 0$ for some positive integer $\displaystyle n$.

Now, how to show that $\displaystyle A^{n+1} \neq 0$???
$\displaystyle A^{n+1}=A^{n}A^{1}$ but, even though $\displaystyle A^{n}, A^{1}$ nonzero matrix, $\displaystyle A^{n+1}$ can be zero matrix also.

I'm stuck here.
What is the determinant of the zero matrix? Can the zero matrix be included in the set of invertible matrices?

3. Originally Posted by Prove It
What is the determinant of the zero matrix? Can the zero matrix be included in the set of invertible matrices?
the proof doesn't need determinant or adjoint.

4. Originally Posted by deniselim17
Let $\displaystyle A \in M_{n}(mathbb{R})$.
If $\displaystyle A$ is invertible, show that $\displaystyle A^{n} \neq 0$ for all positive integer $\displaystyle n$.

I started with $\displaystyle n=1$, then $\displaystyle A^{n}=A^{1} \neq 0$ since $\displaystyle A$ is invertible.

Assume that $\displaystyle A^{n} \neq 0$ for some positive integer $\displaystyle n$.

Now, how to show that $\displaystyle A^{n+1} \neq 0$???
$\displaystyle A^{n+1}=A^{n}A^{1}$ but, even though $\displaystyle A^{n}, A^{1}$ nonzero matrix, $\displaystyle A^{n+1}$ can be zero matrix also.

I'm stuck here.
If $\displaystyle A^{n+1}= 0$ and A is invertible, then $\displaystyle A^{-1}(A^{n+1})$= ?

5. Originally Posted by deniselim17
the proof doesn't need determinant or adjoint.
What I am saying is, that if you check your original definitions, you can't include the zero matrix because its determinant is zero. In other words, the zero matrix is NOT invertible.

6. Originally Posted by Prove It
What I am saying is, that if you check your original definitions, you can't include the zero matrix because its determinant is zero. In other words, the zero matrix is NOT invertible.
Ok. I got it.