Let A be a 2 × 2 real matrix with detA = 1 and trace A = 3. What is the value of trace A2?

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  1. 2
  2. 10
  3. 9
  4. 7

Answer (Detailed Solution Below)

Option 4 : 7
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Detailed Solution

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Calculation: 

Let a and b are two eigen values of A .

a + b = 3   . . . . .  . 1

ab = 1  . . . . .  . . 2

(a - b )2 = (a +b)2 - 4ab 

(a-b ) 2 = 5 

a - b =    . . . .  3 

now from equation 1 and 3 , we get 

a =   and  b =  

trace ( A) = a2 + b2 = 7 

Hence option (4) is correct

Alternate Method

Method -I Let λ1 and λ2 be two eigen values of A. 

Then det A = λ1 λ2 = 1

& Trace A = λ1 + λ2 = 3

And, Eigen values of A2 are λ12, λ22.

⇒ Trace (A2) = λ1+ λ22

Now, (λ1 + λ2)2 = λ1+ λ2+ 2λ1λ2

 (3)2 = λ12 + λ22 + 2

 λ1+ λ2= 9 - 2 = 7

Hence option (4) is correct

Method - II

ChA (x) = x2 - Trace (A) x + Det (A) = 0 

 x2 - 3x + 1 = 0  

 λ2 which are eigenvalues of A .

then eigenvalues of A2 are,

Then trace  = 7

Hence option (4) is correct

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