Let A= bmatrix1 & 2\ 1 & bmatrix and B= bmatrix3 & 3\ & 2 bmatrix. If A2-4A+I=0 and B2-5B-6I=0, then among the two statements: (S1) : [(B-A)(B+A) ]T= bmatrix13 & 15\ 7 & 10 bmatrix and (S2) : (adj(A+B))=-5
- only (S1) is correct
- only (S2) is correct
- both (S1) and (S2) are correct
- both (S1) and (S2) are wrong
Answer
(2) only (S2) is correct
Solution
Every 2 2 matrix M satisfies M2-(tr M)M+|M| I=0 (Cayley–Hamilton).
For A: tr A=1+ =4 and |A|= -2=1 =3
For B: tr B=5 ✓ and |B|=6-3 =-6 =4
∴ A= bmatrix1&2\\1&3 bmatrix, B= bmatrix3&3\\4&2 bmatrix
(S1): B-A= bmatrix2&1\\3&-1 bmatrix, B+A= bmatrix4&5\\5&5 bmatrix
(B-A)(B+A)= bmatrix8+5&10+5\\12-5&15-5 bmatrix= bmatrix13&15\\7&10 bmatrix
[(B-A)(B+A) ]T= bmatrix13&7\\15&10 bmatrix bmatrix13&15\\7&10 bmatrix ∴ (S1) is wrong.
(S2): |A+B|= vmatrix4&5\\5&5 vmatrix=20-25=-5
For a matrix of order n, |adj M|=|M|n-1; here n=2
(adj(A+B))=(-5)1=-5 ∴ (S2) is correct.
Answer: (2)