Math Help - Determinant of a matrix

1. Determinant of a matrix

Hello everyone,
$\begin{vmatrix}x&-1&0&\cdots &0&0\\0&x&-1&\codts &0&0\\0&0&x&\ddots&\vdots&\vdots\\ \vdots&\vdot&\vdots &\ddots&-1&0\\0&0&0&\cdots&x&-1\\a_n&a_{n-1}&a_{n-2}&\cdots&a_1&a_0\end{vmatrix}$

I've tried to solve by myself, but I coudln't get anywhere. I do wish to improve my skill in mathematics; I am not asking for the complete answer. Just give me what to start, then I will try again. I do appreciate your help. Thank you.

2. Re: Determinant of a matrix

$D_2=\begin{vmatrix}{x}&{-1}\\{a_1}&{a_0}\end{vmatrix}=a_0x+a_1$

$D_3=\begin{vmatrix}{x}&{-1}&{0}\\{0}&{x}&{-1}\\{a_2}&{a_1}&{a_0}\end{vmatrix}=xD_2+a_2\begin{ vmatrix}{-1}&{\;\;0}\\{\;\;x}&{-1}\end{vmatrix}=a_0x^2+a_1x+a_2$

Conjecture $D_{n+1}=a_0x^n+a_1x^{n-1}+\ldots +a_n$ and use induction. The way we have found $D_3$ will help you.

3. Re: Determinant of a matrix

Thank you very much!
Assuming $D_{k+1} = a_0x^k+a_1x^{k-1}+\cdots +a_k$,
$D_{k+2}=xD_{k+1}+(-1)^{k+3}a_{k+1}\begin{vmatrix}-1&0&\cdots&\cdots&0\\x&-1&0&\cdots&0\\ \vdots&\vdots&\ddots&\ddots&0\\0&\cdots&\cdots&x&-1\end{vmatrix}$.

and as $\begin{vmatrix}-1&0&\cdots&\cdots&0\\x&-1&0&\cdots&0\\ \vdots&\vdots&\ddots&\ddots&0\\0&\cdots&\cdots&x&-1\end{vmatrix}=(-1)^{k+1}$, we get

$D_{k+2}=xD_{k+1}+a_{k+1}$.

Therefore

$D_{k+2}=a_0^{k+1}+a_1x^{k}\cdots+a_{k+1}$

$D_2$ satisfies the first assumption, therefore via mathematical induction, $D_{n+1}=a_0x^n+a_1^{n-1}+\cdots+a_n\ \ \ ^\forall n\in \mathbb{Z},n\geq 0.$

Am I correct? Thank you very much for your help, and I will work even harder.

4. Re: Determinant of a matrix

Originally Posted by joll
Am I correct?
Yes, you are. Only one thing: it should be $n\geq 1$ instead of $n\geq 0$ . If $n=0$, we have $D_{n+1}=D_1=\det [x]=x$ i.e. we have no $a_i$ and the rule defining the given determinant has no sense.

Thank you very much for your help,
You are welcome!

and I will work even harder.
Good. I am pretty sure.

5. Re: Determinant of a matrix

Thank you very much.

6. Re: Determinant of a matrix

Originally Posted by joll
Hello everyone,
$\begin{vmatrix}x&-1&0&\cdots &0&0\\0&x&-1&\codts &0&0\\0&0&x&\ddots&\vdots&\vdots\\ \vdots&\vdot&\vdots &\ddots&-1&0\\0&0&0&\cdots&x&-1\\a_n&a_{n-1}&a_{n-2}&\cdots&a_1&a_0\end{vmatrix}$

I've tried to solve by myself, but I coudln't get anywhere. I do wish to improve my skill in mathematics; I am not asking for the complete answer. Just give me what to start, then I will try again. I do appreciate your help. Thank you.
it can also be done without using induction. just expand the determinant along the last row and you'll quickly get the answer.

7. Re: Determinant of a matrix

I see... that way, the determinant can be written

$(-1)^{n+1}a_n\begin{vmatrix}-1&0&\cdots&0&0\\x&-1&\cdots&0&0\\0&x&\ddots&\vdots&\vdots\\ \vdots&&\ddots&-1&0\\0&0&\cdots&x&-1\end{vmatrix}+ (-1)^{n+2}a_{n-1}\begin{vmatrix} x&0&\cdots&0&0\\0&-1&\cdots&0&0\\0&x&\ddots&\vdots&\vdots\\ \vdots &&\ddots&-1&0\\0&0&\cdots&x&-1\end{vmatrix}\dots$.

The determinent term in (n,i)cofactor (I mean, the \vmatrix part of the equation above) of the matrix is given by

$(-1)^{n-i}x^{i-1}$

Therefore

$D_{n+1}=a_0x^n+\dots a_n$(am I correct?).

I haven't been learning linear algebra for long yet, so knowing many ways to solve one problem really helps me. Thank you very much.

8. Re: Determinant of a matrix

Originally Posted by joll
I see... that way, the determinant can be written

$(-1)^{n+1}a_n\begin{vmatrix}-1&0&\cdots&0&0\\x&-1&\cdots&0&0\\0&x&\ddots&\vdots&\vdots\\ \vdots&&\ddots&-1&0\\0&0&\cdots&x&-1\end{vmatrix}+ (-1)^{n+2}a_{n-1}\begin{vmatrix} x&0&\cdots&0&0\\0&-1&\cdots&0&0\\0&x&\ddots&\vdots&\vdots\\ \vdots &&\ddots&-1&0\\0&0&\cdots&x&-1\end{vmatrix}\dots$.

The determinent term in (n,i)cofactor (I mean, the \vmatrix part of the equation above) of the matrix is given by

$(-1)^{n-i}x^{i-1}$

Therefore

$D_{n+1}=a_0x^n+\dots a_n$(am I correct?).

I haven't been learning linear algebra for long yet, so knowing many ways to solve one problem really helps me. Thank you very much.
yes, you've got the idea but you forgot that the original determinant is $(n+1) \times (n+1)$ not $n \times n$.

so, after expanding along the last row, you'll get that the determinant is equal to

$\sum_{i=1}^{n+1}(-1)^{n+1+i}a_{n+1-i}(-1)^{n+1-i}x^{i-1}=\sum_{i=1}^{n+1}a_{n+1-i}x^{i-1}=a_n + \ldots + a_0x^n.$

9. Re: Determinant of a matrix

Ah..yes, I forgot that. Thank you.