MATH 2211 Midterm: Exam3210Sample4Ans
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An n n matrix a is called skew-symmetric if at = a. Show that if a is skew-symmetric and n is an odd positive integer, then a is not invertible. By the properties of determinant, det(at) = det( a) det(a) = det( a) det(a) = ( 1)ndet(a) det(a) = det(a). So, we get det(a) = 0 which implies that a is not invertible. Every row of a is multiplied by -1. Solve the linear system using cramer"s rule: x1 + 2x2 + 3x3 = 6. 2x2 + 3x3 = 5 x3 = 1. Answer x1 = x2 = x3 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Therefore, (1, 1, 1) is the solution of the given system. = 1. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) De ne the linear transformations t : p2 p2 by. T(ax2 + bx + c) = ax2 + (a + b)x + (a + b + c). a. )
