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

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 ( A² ) = 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) = λ12 + λ22

Now, (λ₁ + λ₂)² = λ12 + λ22 + 2λ₁λ₂

⇒ (3)² = λ₁² + λ₂² + 2

⇒ λ12 + λ22 = 9 - 2 = 7

Hence option (4) is correct

Method - II

Ch_A (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