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Question

The product of the non-zero eigen values of the matrix ⎢ ⎢ ⎢ ⎢ ⎢ ⎢1000101110011100111010001⎥ ⎥ ⎥ ⎥ ⎥ ⎥ is_____.
  1. 6

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Solution

The correct option is A 6
Let the given matrix be 'A'
Characteristic Equation is |AλI|=0
∣ ∣ ∣ ∣ ∣1λ000101λ110011λ100111λ010001λ∣ ∣ ∣ ∣ ∣=0

Expanding by 1st row
(1λ)∣ ∣ ∣ ∣1λ11011λ10111λ00001λ∣ ∣ ∣ ∣+∣ ∣ ∣ ∣01λ11011λ10111λ1000∣ ∣ ∣ ∣=0
or (1λ)2∣ ∣1λ1111λ1111λ∣ ∣(1)∣ ∣1λ1111λ1111λ∣ ∣=0 ...(1)

Let B=1λ1111λ1111λ

|B|=λ2(λ3)

So (1λ)2|B|(1)|B|=0 [using (1)]

|B|[(1λ)21]=0

λ2(λ3)[λ(λ2)]=0
λ3(λ2)(λ3)=0
So λ=0,0,0,2,3
Hence required product =2×3=6

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