# Calculate the determinant of $n^\text{th}$ order

Calculate the determinant of $$n^\text{th}$$ order: $$\begin{vmatrix} 1 + a_1 & 1 + a_1^2 & \dots & 1 + a_1^n \\ 1 + a_2 & 1 + a_2^2 & \dots & 1 + a_2^n \\ \vdots & \vdots & \ddots & \vdots \\ 1 + a_n & 1 + a_n^2 & \dots & 1 + a_n^n \\ \end{vmatrix}$$

So whenever any two of the variables are equal the determinant becomes $$0$$. Therefore, it has $$\prod_{1 \le k < i \le n} (a_i - a_k)$$ as a factor. But I haven't been able to find the rest of the factors.

Any help is appreciated.

• Since the determinant has degree $n(n+1)/2$ but your known factor has degree $n(n-1)/2$, you need to find $n$ more linear factors. Work them out for low $n$ to form a conjecture. – J.G. Jul 13 at 14:26

When after experimentation we find that the determinant is a product of some factors, it's a good idea to see if the matrix itself can be factored into a product of matrices. If the matrix $$A = BC$$, where the determinants of $$B$$ and $$C$$ are easily calculated, then we can recover $$\det(A)$$ using the identity $$\det (BC) = \det (B) \det (C)$$.

Here, as you noted, $$\prod_{1 \leq i < j \leq n} (a_j - a_i)$$ is a factor of the final determinant. It is well-known that this is the determinant of the Vandermonde matrix $$V = \begin{bmatrix} 1 & a_1 & a_1^2 & \cdots & a_1^{n-1} \\ 1 & a_2 & a_2^2 & \cdots & a_2^{n-1} \\ 1 & a_3 & a_3^2 & \cdots & a_3^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & a_n & a_n^2 & \cdots & a_n^{n-1} \end{bmatrix}$$ so this begs of whether we can factor the given matrix $$A$$ as $$A = V B$$ for some matrix $$B$$. And in fact, we can! In fact, we can show that with $$B = \begin{bmatrix} 1 & 1 & 1 & \cdots & 1& (-1)^{n-1} e_n + 1 \\ 1 & 0 & 0 & \cdots & 0& (-1)^{n-2} e_{n-1} \\ 0 & 1 & 0 & \cdots & 0 & (-1)^{n-3} e_{n-2} \\ \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ \vdots & \vdots &\vdots & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & (-1)^0 e_1\end{bmatrix}$$ the factorization $$A = V B$$ holds. Here the $$e_i$$ denotes the degree $$i$$ elementary symmetric polynomial in the variables $$a_1, \cdots a_n$$. The correctness of the first $$n - 1$$ columns of $$A$$ resulting from this product are easy to verify. What about the pesky last column? We basically want to verify that $$V \cdot \begin{bmatrix} (-1)^{n-1} e_n \\ (-1)^{n-2} e_{n-1} \\ \vdots \\ e_1 \end{bmatrix} = \begin{bmatrix} a_1^n \\ a_2^n \\ \vdots \\ a_n^n\end{bmatrix}$$ because if this is true, then the last column will also match, since by adding the extra $$+1$$ in the upper right entry of $$B$$ the last column will be $$V \cdot \left(~\begin{bmatrix} (-1)^{n-1} e_n \\ (-1)^{n-2} e_{n-1} \\ \vdots \\ e_1 \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}~\right) = V \cdot \begin{bmatrix} (-1)^{n-1} e_n \\ (-1)^{n-2} e_{n-1} \\ \vdots \\ e_1 \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix} = \begin{bmatrix} 1 + a_1^n \\ 1+ a_2^n \\ \vdots \\ 1 + a_n^n \end{bmatrix}$$

Note that this is equivalent to asserting

Lemma: For all of the $$a_i$$, the following holds: $$a_i^n = \sum_{k = 0}^{n-1} (-1)^{n - k - 1} \cdot a_i^k \cdot e_{n - k}$$

Proof of Lemma: This fact follows because the elementary symmetric polynomials are the resulting coefficients of the monic polynomial $$p(\lambda) = \prod_{i = 1}^n (\lambda - a_i) = \lambda^n - e_1 \lambda^{n-1} + e_2 \lambda^{n-2} - \cdots + (-1)^n e_n$$ Plugging in $$\lambda = a_i$$, we have $$p(a_i) = \sum_{k = 0}^n (-1)^{n-k} \cdot a_i^k \cdot e_{n-k} = 0 \implies a_i^n = \sum_{k = 0}^{n-1} (-1)^{n-k - 1} \cdot a_i^k \cdot e_{n - k}$$

We've now verified our factorization is correct, so what remains is to calculate the determinant of $$B$$. Laplace expanding along the last column, we can see that \begin{align*} \det B = (-1)^{n+1} [(-1)^{n-1} \cdot e_n + e_0] &+ \sum_{k = 1}^{n-1} \left[(-1)^{n + k - 1} \cdot \det(B_{k, n}) \cdot (-1)^{n - k - 1} \cdot e_{n-k}\right] \\ \\ &= e_n + (-1)^{n-1} \cdot e_0 + \sum_{k = 1}^{n-1} \det(B_{k, n}) \cdot e_{n - k} \end{align*} where $$B_{k, n}$$ is the minor $$B$$ obtained by removing the $$k$$th row and $$n$$th column. It is not too difficult to verify that $$\det(B_{k, n}) = (-1)^{k - 1}$$ (although if it is not clear, I can post an addendum explaining why). Hence, the determinant of $$B$$ is $$e_n + (-1)^{n + 1} e_0 + \sum_{k = 1}^{n-1} \det(B_{k, n}) \cdot e_{n - k} = e_n + e_{n-1} - e_{n-2} + \cdots + (-1)^{n + 1} e_0$$ So it follows that $$\det A = \det(V) \det (B) = \left[\prod_{1 \leq i < j \leq n} (a_j - a_i)\right] \cdot (e_n + e_{n - 1} - e_{n-2} + \cdots + (-1)^{n+1} e_0)$$ verifying lhf's conjecture. (Note that the above expression also has the correct sign as well). $$\square$$

• Thank you for your detailed answer! – VIVID Jul 14 at 3:46
• I think one should have $i \neq j$ as the index of the Vandermonde product, otherwise will get $0$ – Azamat Abdullaev Jul 14 at 5:06
• @NodeJS Fixed; thanks for catching that! – paulinho Jul 14 at 14:09

Partial answer: (with the aid of WA)

For $$n=1$$, the other factor is $$e_1+e_0$$.

For $$n=2$$, the other factor is $$e_2+e_1-e_0$$.

For $$n=3$$, the other factor is $$e_3+e_2-e_1+e_0$$.

For $$n=4$$, the other factor is $$e_4+e_3-e_2+e_1-e_0$$.

Conjecture:

The other factor is $$e_n+e_{n-1}-e_{n-2}+e_{n-3}- \cdots - (-1)^n e_0$$

Here, $$e_i$$ are the elementary symmetric polynomials.

Also, perhaps this is only up to sign.

• Well, this seems to be correct. But I've been trying to prove this for half an hour now :) – VIVID Jul 13 at 15:22
• @VIVID, the matrix determinant lemma is probably useful here but I haven't tried. – lhf Jul 13 at 20:02
• @lhf I think your conjecture is right, even up to the sign (so long as the term $(a_i - a_j)$ appears iff $i < j$)! Please see my answer. – paulinho Jul 13 at 21:50