How many involutary matrices of order 2013 and trace = 2013 exist having a57 = -1?
My Attempt: An involutary matrix is the one which satisfies : $A^2$ = I . Now, if k is an eigen value of A, then it satisfies : $k^2$ - 1 =0 which gives the possible eigen values of the involutary matrix A as 1 or -1 .
Now, for a matrix of order 2013, trace = 2013 clearly means that all it's eigen values are 1.
Now. i understand that had the condition a57 = -1 NOT been given , the IDENTITY matrix is one of the matrices satisfying the above conditions.
BUT how do i prove that the identity matrix is the only involutary matrix of order 2013 which has it's trace= 2013? Thank you.
(this question came up before diagonalization came up in the course which i am reading which i think might be early for the question to crop up :) )