Let B= [ array ll 1 & 3 1 & 5 array ] and A be a 2 2 matrix such that A B⁻¹=A⁻¹ . If B C B⁻¹=A and C^4+ C^2+ I=O , then 2 - is equal to

Step-by-step solution

aligned & BCB ⁻¹= A & ( BCB ⁻¹ ) ( BCB ⁻¹ )= A A & BCI CB ⁻¹= A ^2 & BC ^2 ~B ⁻¹= A ^2 & B ⁻¹ ( BC ^2 ~B ⁻¹ ) B = B ⁻¹ (A.A)B aligned From equation (1) aligned & C ^2= A ⁻¹ A B & C ^2= B aligned aligned & Also AB ⁻¹= A ⁻¹ & AB ⁻¹ A = A ⁻¹ ~A = I & A ⁻¹ ( AB ⁻¹ ~A )= A ⁻¹ I & B ⁻¹ ~A = A ⁻¹ aligned Now characteristics equation of C ^2 is aligned & |C₂- I |=0 & |B- I|=0 aligned aligned & | array cc 1- & 3 1 & 5- array |=0 & (1- )(5-1)-3=0 ( ^2-6 +5 )-3=0 & ^2-6 +2=0 & ^2-6 ~B +2 I =0 & C ^4-6 C ^2+2 I =0 & =-6 & =2 &