Determinant of a square matrix with a particular pattern

Question

Let $A=[a_{ij}]$ be the square matrix of order $n$ whose entries are given as follows. For $1\le i,j\le n$ we have $$a_{ij}= \begin{cases} ij,&\text{if $i=j$}\\ ij+1,&\text{if $i\neq j$} \end {cases}$$ Evaluate the determinant of $A$.

I took an arbitrary $3\times 3$ and then an arbitrary $4\times 4$ matrix to see what things go like. Well the matrices do have an interesting pattern :

They look like a sequence of $L_i$ and $L_i'$ joined at right angle with decreasing lenghts where $L_i$ is the $i$-th row without $i-1$ elements and $L_i'$ is the $i$-th coloumn without $i-1$ elements.

But the requirement is to find the determinant . May be the pattern could contribute to that , I guessed. PLease give me some clues as to how to work out this problem .

Thanks for any help.

Answer 1

The matrix is in the form of $A=vv^T+nuu^T-I$, where $v^T=(1,2,\ldots,n)$ and $u^T=\frac1{\sqrt{n}}(1,1,\ldots,1)^T$. We can rewrite $v$ as the sum of two mutually orthogonal components $x=(v,u)u=\frac{(n+1)\sqrt{n}}{2}u$ and $y=v-(v,u)u=v-\frac{n+1}2(1,1,\ldots,1)^T$. Therefore $A$ is orthogonally similar to $$ \pmatrix{\|x\|^2+n-1&\|x\|\|y\|\\ \|x\|\|y\|&\|y\|^2-1}\oplus (-I_{n-2}) $$ and hence \begin{align*} \det A &=\left[(\|x\|^2+n-1)(\|y\|^2-1)-\|x\|^2\|y\|^2\right](-1)^{n-2}\\ &=(-1)^n\left[(n-1)(\|y\|^2-1)-\|x\|^2\right]\\ &=(-1)^n\left[(n-1)\left(\sum_{k=1}^n\left(k-\frac{n+1}2\right)^2-1\right) -\frac{n(n+1)^2}4\right]\\ &=\frac{(-1)^n}{12} (n^4 - 4n^3 - 7n^2 - 14n + 12). \end{align*}

Answer 2

The first $20$ determinants, calculated with PARI/GP

? for(n=1,20,a=matrix(n,n,x,y,0);for(i=1,n,for(j=1,n,if(i<>j,a[i,j]=i*j+1);if(i=
=j,a[i,j]=i*j)));print(n,"  ",matdet(a),"    ",factor(matdet(a))))
1  1    matrix(0,2)
2  -5    [-1, 1; 5, 1]
3  10    [2, 1; 5, 1]
4  -13    [-1, 1; 13, 1]
5  9    Mat([3, 2])
6  9    Mat([3, 2])
7  -50    [-1, 1; 2, 1; 5, 2]
8  125    Mat([5, 3])
9  -247    [-1, 1; 13, 1; 19, 1]
10  431    Mat([431, 1])
11  -694    [-1, 1; 2, 1; 347, 1]
12  1055    [5, 1; 211, 1]
13  -1535    [-1, 1; 5, 1; 307, 1]
14  2157    [3, 1; 719, 1]
15  -2946    [-1, 1; 2, 1; 3, 1; 491, 1]
16  3929    Mat([3929, 1])
17  -5135    [-1, 1; 5, 1; 13, 1; 79, 1]
18  6595    [5, 1; 1319, 1]
19  -8342    [-1, 1; 2, 1; 43, 1; 97, 1]
20  10411    [29, 1; 359, 1]
?

There does not seem to be any pattern. I added the factorizations, but I cannot notice any regularity. It is strange that the sign changes always with the exception $5->6$ upto $n=100$