Let A be a 3 3 real matrix such that X^T A X = O for all non-zero 3 1 real matrices X . The elements of the first row of A form an arithmetic progression with a positive common difference, and the elements of the second row of A also form an arithmetic progression. If the sum of the squares of all elements of A is 108 , then the value of (A^2 + 60I) is

Step-by-step solution

Since X^T A X = O for all 3 1 matrices X , A is a skew-symmetric matrix. Let A = bmatrix 0 & x & y -x & 0 & z -y & -z & 0 bmatrix . The first row is 0, x, y . Since these form an arithmetic progression with a positive common difference, we have: x - 0 = y - x y = 2x , and x > 0 . The second row is -x, 0, z . Since these also form an arithmetic progression, we have: 0 - (-x) = z - 0 z = x . The sum of the squares of all elements of A is given as 108 . 0^2 + x^2 + y^2 + (-x)^2 + 0^2 + z^2 + (-y)^2 + (-z)^2 + 0^2 = 10