Let A = bmatrix 1 & 2 1 & bmatrix and B = bmatrix 3 & 3 & 2 bmatrix . If A^2 - 4A + I = O and B^2 - 5B - 6I = O , then among the two statements : (S1): [(B-A)(B+A)]^T = bmatrix 13 & 15 7 & 10 bmatrix and (S2): ( adj (A+B)) = -5 ,

Step-by-step solution

For a 2 2 matrix M , the characteristic equation is given by M^2 - Tr (M)M + (M)I = O . For matrix A = bmatrix 1 & 2 1 & bmatrix , we are given A^2 - 4A + I = O . Comparing the trace, we get Tr (A) = 1 + = 4 = 3 . Thus, A = bmatrix 1 & 2 1 & 3 bmatrix . For matrix B = bmatrix 3 & 3 & 2 bmatrix , we are given B^2 - 5B - 6I = O . Comparing the determinant, we get (B) = 6 - 3 = -6 3 = 12 = 4 . Thus, B = bmatrix 3 & 3 4 & 2 bmatrix . Evaluating Statement (S1): B - A = bmatrix 3 & 3 4 & 2 bmatrix - bmatrix 1 & 2 1 & 3 b