Let A be a 3 3 real matrix such that X^T A X = O for all non-zero 3 1 real matrices X . If A bmatrix 1 1 1 bmatrix = bmatrix 1 1 -2 bmatrix and A bmatrix 1 -1 0 bmatrix = bmatrix -2 -2 4 bmatrix , then the value of (A^4 + 15A^2 + 16I) is

Step-by-step solution

Since X^T A X = O for all 3 1 matrices X , A must be a skew-symmetric matrix. Let A = bmatrix 0 & x & y -x & 0 & z -y & -z & 0 bmatrix . Given A bmatrix 1 1 1 bmatrix = bmatrix 1 1 -2 bmatrix , we get: x + y = 1 -x + z = 1 -y - z = -2 y + z = 2 Given A bmatrix 1 -1 0 bmatrix = bmatrix -2 -2 4 bmatrix , we get: -x = -2 x = 2 -x = -2 x = 2 -y + z = 4 Using x = 2 in x + y = 1 , we get y = -1 . Using x = 2 in -x + z = 1 , we get z = 3 . These values satisfy all equations. The characteristic equation of a 3 3 skew-symme