Error in calculating determinant

Question

Calculate n-order determinant (n>1):

$\left|\begin{array}{cccccccc}a & b & 0 & 0 & \cdots & 0 & 0 & 0 \\ 0 & a & b & 0 & \cdots & 0 & 0 & 0 \\ 0 & 0 & a & b & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 0 & a & b \\ b & 0 & 0 & 0 & \cdots & 0 & 0 & a\end{array}\right|$

The result should be

$a^{n}+(-1)^{n+1} b^{n}$

I can use the MMA code to get this result:

Clear["Global`*"];
s[n_] := SparseArray[{{n, 1} -> b, {i_, j_} /; j - i == 1 -> 
    b, {i_, j_} /; j == i -> a}, {n, n}]
detmat[n_] := Det[s[n]]
tab = Table[detmat[i], {i, 2, 10}]

$\left\{a^{2}-b^{2}, a^{3}+b^{3}, a^{4}-b^{4}, a^{5}+b^{5}, a^{6}-b^{6}, a^{7}+b^{7}, a^{8}-b^{8}, a^{9}+b^{9}, a^{10}-b^{10}\right\}$

FindSequenceFunction[tab, n - 1]

$a^{n}+(-b)^{-1+n} b$

However, when I tried another method, I made a mistake. How should I modify this code?

Clear["Global`*"];
detNewMatrix[n_] := 
 Det[Module[{i = 1, j = 1, mm = Array[m, {n, n}]}, 
   For[i = 1, i <= n, i++, 
    For[j = 1, j <= n, j++, 
     If[j == i + 1, m[i, j] = b, 
      If [i == n && j == 1, m[i, j] = b, 
       If[i == j, m[i, j] = a, m[i, j] = 0]]]]]; mm]]
tab = Table[detNewMatrix[k], {k, 2, 10}]

$\left\{a^{2}-b^{2}, a^{3}-a b^{2}+b^{3}, a^{4}+a b^{3}-b^{4}, a^{5}-a b^{4}+b^{5}, a^{6}+a b^{5}-b^{6}, a^{7}-a b^{6}+b^{7}, a^{8}+a b^{7}-b^{8}, a^{9}-a b^{8}+b^{9}, a^{10}+a b^{9}-b^{10}\right\}$

I think there may be a "coverage" in the code. It is correct to directly calculate any determinant, but when listing together, only the first determinant is calculated correctly, and the subsequent determinants are calculated incorrectly. But I don't know what went wrong.

Answer 1

Change mm[i,j]to mm[[i,j]] in your last approach!

newmatrix[n_] := 
 Det@Module[{i = 1, j = 1, mm = Array[m, {n, n}]}, 
   For[i = 1, i <= n, i++, For[j = 1, j <= n, j++,
     If[j == i + 1, mm[[i, j]] = b,
      If[i == n && j == 1, mm[[i, j]] = b,
       If[i == j, mm[[i, j]] = a, mm[[i, j]] = 0]]]]]; mm] 

newmatrix[6] (*a^6-b^6*)

Answer 2

• Band is another approach.

table = Table[
  Det@SparseArray[{Band[{1, 1}] -> a, Band[{1, 2}] -> b, 
     Band[{n, 1}] -> b}, n], {n, 2, 10}]
FindSequenceFunction[table, n - 1]

{a^2 - b^2, a^3 + b^3, a^4 - b^4, a^5 + b^5, a^6 - b^6, a^7 + b^7, a^8 - b^8, a^9 + b^9, a^10 - b^10}

a^n + (-b)^(-1 + n) b

• RotateRight+ PadRight.

Table[Det@NestList[RotateRight, PadRight[{a, b}, n + 1], n], {n, 1, 
  10}]

{a^2 - b^2, a^3 + b^3, a^4 - b^4, a^5 + b^5, a^6 - b^6, a^7 + b^7, a^8 - b^8, a^9 + b^9, a^10 - b^10, a^11 + b^11}