Maths Ex XXXXXXXXXX

4) An is an n X n matrix
aii = x
aij = 1 if j =i+1 or i-1
dn = det An
An+2 =
[












]





det An+2 = ∑


(where is the minor = determinant of A with i
th row and jth column
chucked out)
Expanding about 1st row:
det An+2 = x*(M11) – 1*(M12) = x*(M11) – M12
but M11 = det(An+1) (since after chucking out 1
st row and 1st column we have an An+1)
M12 = det(
[










]





)
det An+1= ∑



Expanding about 1st column:
M12 = 1*Mn+1,n+1 = Mn+1,n+1 = det(An) = dn (Mn+1,n+1 is the cofactor of An+1)
 det An+2 = x*(M11) – 1*(M12) = x*(M11) – M12 = x*( det(An+1)) -...