SOLUTION: How would you find the determinant of the n x n matrix: a b 0 0...0 0 a b 0...0 0 0 a b...0 ........... 0 0 0 0...b b 0 0 0...a I understand that i need to use laplace exp

Algebra ->  Matrices-and-determiminant -> SOLUTION: How would you find the determinant of the n x n matrix: a b 0 0...0 0 a b 0...0 0 0 a b...0 ........... 0 0 0 0...b b 0 0 0...a I understand that i need to use laplace exp      Log On


   

Question 243802: How would you find the determinant of the n x n matrix:
a b 0 0...0
0 a b 0...0
0 0 a b...0
...........
0 0 0 0...b
b 0 0 0...a
I understand that i need to use laplace expansion but this problem is confusing me just because you dont know the numbers in the matrix or exactly how many rows and columns you are working with
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If we look at the 2x2 matrix
a b
b a

we can clearly see that the determinant is a%5E2-b%5E2 (just use the definition of the 2x2 determinant).


Moving onto the 3x3 matrix

a b 0
0 a b
b 0 a

If we compute this determinant, we get




Similarly, for the 4x4 matrix

a b 0 0
0 a b 0
0 0 a b
b 0 0 a



If we apply the laplace expansion along the first column, we get . Also, for the 5x5 matrix

a b 0 0 0
0 a b 0 0
0 0 a b 0
0 0 0 a b
b 0 0 0 a


the determinant is a%5E5%2Bb%5E5. Are you starting to see a pattern? If not, then try to find the determinant of the 6x6 matrix

a b 0 0 0 0
0 a b 0 0 0
0 0 a b 0 0
0 0 0 a b 0
0 0 0 0 a b
b 0 0 0 0 a


You'll find that the determinant is a%5E6-b%5E6


In general, the determinant of the n x n matrix


a b 0 0...0
0 a b 0...0
0 0 a b...0
...........
0 0 0 0...b
b 0 0 0...a


is a%5En%2B%28-1%29%5E%28n%2B1%29b%5En


I left some work out due to time constraints. Let me know if you need to see how to get any determinants.