Let A be a 3 × 3 real matrix such that A 1 1 0 = 1 1 0 ; A 1 0 1 = - 1 0 1 and A 0 0 1 = 1 1 2 . If X = x 1 x 2 x 3 T and I is an identity matrix of order 3 , then the system A - 2 I X = 4 1 1 has

Step-by-step solution

Let the matrix be A = a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 Now A 0 0 1 = c 1 c 2 c 3 = 1 1 2 ⇒    c 1 = 1 , c 2 = 1 , c 3 = 2 A 1 0 1 = c 1 + a 1 c 2 + a 2 c 3 + a 3 = - 1 0 1 ⇒    a 1 = - 2 , a 2 = - 1 , a 3 = - 1 A 1 1 0 = a 1 + b 1 a 2 + b 2 a 3 + b 3 = 1 1 0 ⇒    b 1 = 3 , b 2 = 2 , b 3 = 1 So matrix A = - 2 3 1 - 1 2 1 - 1 1 2 ⇒ A - 2 I = - 4 3 1 - 1 0 1 - 1 1 0 i.e. A - 2   I = 0 Now X = x 1 x 2 x 3 So,    - 4 3 1 - 1 0 1 - 1 1 0 x 1 x 2 x 3