Maths Ex XXXXXXXXXX

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4* Let An be the n x n matrix given by (la = aii=lifj=it 1 ash = 0 (all other i, j)
Let 4 = det An. By expanding, show that 4+2 = x4+1 - 4-x = 2, prove by induction that 4 = n + 1.
5* Let An be the n x n matrix given by
au = x
sij= 1 (i
In the case
By using row/column operations and expansions, find det(An). For which values of x is An invertible?

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Maths Ex 10 - 32013

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Answered Same DayDec 22, 2021

Answer To: Maths Ex XXXXXXXXXX

David answered on Dec 22 2021

115 Votes

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)) -...

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Maths Ex XXXXXXXXXX

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