Let I be the identity matrix of order 3 3 and for the matrix A = [ array ccc & 2 & 3 4 & 5 & 6 7 & -1 & 2 array ],| A |=-1 . Let B be the inverse of the matrix adj ( A adj ( A ^2 ) ) . Then |( B+1)| is equal to _____

Step-by-step solution

aligned & |A|= | array ccc & 2 & 3 4 & 5 & 6 7 & -1 & 2 array |=-1 & (16)-2(-34)+3(-39)=-1 & 16 =48 =3 & ~B ⁻¹= adj ( A adj ( ~A ^2 ) ) & Let C = A adj ( ~A ^2 ) & A C=A^2 adj (A^2 )=|A|^2 I=I C=A⁻¹ & Now B⁻¹= adj (A⁻¹ )=B= adj (A) & Now B + I 3 ~B + I & Let P =3 ~B + I & P =3 adj ( ~A )+ I & AP =3 Aadj ( ~A )+ A & AP =3| ~A | I + A & AP = A -3 I & | AP |=| A -3 I | & | A | | P |= | array ccc 0 & 2 & 3 4 & 2 & 6 7 & -1 & -1 array |=38 & | P |=-38 aligned