<s>[INST] Show that if $a\neq b$, then we have for the $n\times n$-matrix $\textrm{det}=\frac{a^{n+1}-b^{n+1}}{a-b}$. The Problem
Show that if $a\neq b$, then we have for the $n\times n$-matrix
$$\textrm{det}\begin{pmatrix} a+b & ab & 0 & \ldots & 0 & 0 \\
1 & a+b & ab & \ldots & 0 & 0 \\
0 & 1 & a+b & \ldots & 0 & 0 \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
0 & 0 & 0 & \ldots & a+b & ab \\
0 & 0 & 0 & \ldots & 1 & a+b\end{pmatrix} =\frac{a^{n+1}-b^{n+1}}{a-b}$$
What if $a=b$?

My Questions
I am not entirely sure how to begin this proof.  I suspect I am missing something that makes it quite simple.  I tried looking for similar problems, and I noticed a common theme being the use of row operations to rewrite the matrix, from which the determinant was found.  However, I still didn't understand many of the intermediate computations when it came to actually finding the determinant.  My questions are as follows.


*

*Should I use row operations to rewrite this matrix?  If so, what would be an example of how that would look computationally?

*In either case, how should I go about actually computing the determinant to show the statement is true?  Is there an algorithm that is helpful here?

*I noticed for the follow up question that, if $a=b$, then $$\textrm{det}\begin{pmatrix} a+a & aa & 0 & \ldots & 0 & 0 \\
1 & a+a & aa & \ldots & 0 & 0 \\
0 & 1 & a+a & \ldots & 0 & 0 \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
0 & 0 & 0 & \ldots & a+a & aa \\
0 & 0 & 0 & \ldots & 1 & a+a\end{pmatrix}.$$  Am I right in my thinking there?



Other Details
The book used in the course is Abstract Linear Algebra by Curtis.  It has been of little help to me here...
 [/INST] Let's call $A_n$ the matrix for which you want to compute the determinant and let's see how it can be constructed iteratively:
$$ A_{n+1} = \begin{pmatrix} a+b & ab & 0 & \ldots & 0 & 0 \\
1 & a+b & ab & \ldots & 0 & 0 \\
0 & 1 & a+b & \ldots & 0 & 0 \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
0 & 0 & 0 & \ldots & a+b & ab \\
0 & 0 & 0 & \ldots & 1 & a+b\end{pmatrix}
 = \begin{pmatrix}  
 A_n  &  \begin{matrix}  0 \\ \cdots \\ 0 \\ ab  \end{matrix} \\ 
 \begin{matrix} 0 & \cdots &  0 &  1 \end{matrix} & a+b
\end{pmatrix} $$
Such layout cries out for induction to compute the determinant $D_n = \det(A_n)$ . For $n=1$, we have $A_1 = (a+b)$ so $D_1 = a+b$ and $D_1 \cdot (a-b) = a^2 - b^2 $.
Now, if we assume the formula is true up to an integer $n$, can we prove that it is also true for $n+1$ ?
We can use  Laplace expansion applied to the last column to find out that:
$$ D_{n+1} = (a+b)\cdot D_n - ab \cdot D_{n-1}$$
Therefore:
$$ \begin{align} (a-b)\cdot D_{n+1} & = (a+b)\cdot (a-b) D_n - ab \cdot(a-b) D_{n-1} \\
& = (a+b)\cdot( a^{n+1} - b^{n+1} ) - ab \cdot( a^{n} - b^{n} ) \\
& = a^{n+2}- ab^{n+1} + ba^{n+1} - b^{n+2} - ba^{n+1}+  ab^{n+1}  \\
& = a^{n+2}- b^{n+2}\\
\end{align}  $$
And we are done for the first question.
The second question asks what is happening when $a=b$. Obviously we can't apply the formula above since it is only valid for $a \ne b $. But we can take $b$ as close as we want to $a$, let's say $b_k = a + \frac{1}{k}$. Now we can apply the previous result, meaning that for all positive integer $k$:
$$ \det \begin{pmatrix} a+(a + \frac{1}{k}) & a(a + \frac{1}{k})  & \ldots & 0 & 0 \\
1 & a+(a + \frac{1}{k}) & \ldots & 0 & 0 \\
0 & 1 &  \ldots & 0 & 0 \\
\ldots & \ldots & \ldots & \ldots & \ldots \\
0 & 0  & \ldots & a+(a + \frac{1}{k}) & a(a + \frac{1}{k}) \\
0 & 0  & \ldots & 1 & a+(a + \frac{1}{k})\end{pmatrix} = \frac{\left(a + \frac{1}{k}\right)^{n+1} - a^{n+1}}{ \frac{1}{k} }  $$
To conclude, you can use the continuity of the determinant and take the limit on both sides as $k \to \infty $. On the left you have the determinant of the matrix $A_n$ with $a=b$ and on the right you have the derivative of the function $ x \mapsto  x^{n+1} $ at $x=a$, which is $(n+1)a^n$.
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