GATE Electrical Engineering (EE) 2018 Solved Paper

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Question : 42 of 65

Marks: +1, -0

Let

A=[\table 1 , 0 , -1 ; -1 , 2 , 0 ; 0 , 0 , -2]

and

B=A^{3}-A^{2}-4 A+5 I

, where

I

is the

3 × 3

identity matrix. The determinant of

B

is ___________(upto 1 decimal place).

Your Answer:

Solution:

A=[\table 1 , 0 , -1 ; -1 , 2 , 0 ; 0 , 0 , -2]

|A-λ I|=0

|\table 1-λ , 0 , -1 ; -1 , 2-λ , 0 ; 0 , 0 , -2-λ|=0

(1-λ)((2-λ)(-2-λ))-1(0-0)=0

λ=1,2,-2

Eigen values of

A

are

1,2,-2

Eigen values of

A^{3}

are

1,8,-8

A^{2}

are

1,4,4

4 A

are

4,8,-8

5 I

are

5,5,5

A^{3}-A^{2}-4 A+5 I

are

1,1,1

∴ |B|=(1)(1)(1)=1

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