Let A be a 2 × 2 real matrix and I be the identity matrix of order 2 . If the roots of the equation | A - xI | = 0 be - 1 and 3 , then the sum of the diagonal elements of the matrix A 2 is _____.

Step-by-step solution

Given: | A - xI | = 0 have roots - 1 and 3 . Let, A = a b c d ⇒ A - xI = a b c d - x 0 0 x ⇒ A - xI = a - x b c d - x ⇒ A - xI = a - x d - x - bc ⇒ A - xI = ad - ax - dx + x 2 - bc ⇒ A - xI = x 2 - a + d x + ad - bc Sum of roots, a + d = 2 Product of roots, ad - bc = - 3 Now finding matrix A 2 , ⇒ A 2 = a b c d × a b c d ⇒ A 2 = a 2 + bc ab + bd ac + cd bc + d 2 Sum of diagonal elements, S = a 2 + bc + bc + d 2 ⇒ S = a 2 + 2 bc + d 2 ⇒ S = ( a + d ) 2 - 2 ad - bc ⇒ S = ( 2 ) 2 - 2 - 3 ⇒ S = 4 + 6 ⇒ S = 10