Matrices

Let A be a 2 2 symmetric matrix such that A bmatrix1\\1 bmatrix = bmatrix3\\7 bmatrix and the determinant of A be 1. If A-1 = A + I, where I is an identity matrix of order 2 2, then + equals .

Let A=I2-2MMT, where M is a real matrix of order 2 1 such that the relation MTM=I1 holds. If is a real number such that the relation AX= X holds for some non-zero real matrix X of order 2 1, then the sum of squares of all possible values of is equal to .

Let A be a square matrix such that AAT=I. Then 12A [ (A+AT )2+ (A-AT )2 ] is equal to

1. A2+AT
2. A3+I
3. A3+AT
4. A2+I

Consider the matrix f(x)= bmatrix x & - x & 0\ x & x & 0\ 0 & 0 & 1 bmatrix. Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

1. Statement I is false but Statement II is true
2. Both Statement I and Statement II are false
3. Both Statement I and Statement II are true
4. Statement I is true but Statement II is false

Let A be a 2 2 real matrix and I be the identity matrix of order 2. If the roots of the equation |A-xI|=0 be -1 and 3, then the sum of the diagonal elements of the matrix A2 is .