(1) (20 points) Indicate whether the statement is true (T) or false (F) without
ID: 3186328 • Letter: #
Question
(1) (20 points) Indicate whether the statement is true (T) or false (F) without proof. Each right answer is for 2 points and wrong answer is for -1 points. (a) A 4 x 4 orthogonal matrix has always a real cigenvalue. (b) If A is a symmetric matrix such that As = 0,then A-0. (c) The kernel of the differential operator ( 100 on Co (R) has dimension 100. thogonal. space of all polynomials. most n in t with real coeflicients. Then the set W (d) If an orthogonal matrix A is similar to B, then B is also or- (e) If all the cigenvalucs of A are zero, then we have A-0. (f) The transformation T)()f(2) -23f (x) is linear on the (g) Denote by Pn the vector space of polynomials of degree at [p(t) E Pn I p(5) Z) is a subspace of P (h) Singular matrices are not diagonalizable. (i) If A is diagonalizable and has characteristic polynomial p(A) = (A-1)(A - 2)3(A -3)6, then the rank of 31 - A is 4. G) Any square matrix A is the sum of two invertible matrices.
Explanation / Answer
a) The statement is true.
b) The statement is true.
c) The statement is true.
d) The statement is true.
e) The statement is false.
f) The statement is true.
g) The statement is true.
h) The statement is false.
i) The statement is true.
j) The statement is false.
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