ECON 715 Midterm: ECON715Midterm1Fall2018solution
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Instructions: one double-sided sheet with any content is allowed, calculators are not allowed, show all the calculations, and explain your steps, if you need more space, use the back of the page, fully label all graphs. Xn(cid:2)kx 0 k(cid:2)nxn(cid:2)k is k (cid:2) k k(cid:2)n is n (cid:2) n. 2 (c) prove that x 0x and xx 0 are symmetric. A symmetric matrix is equal to its transpose. Thus, (x 0x)0 = x 0 (x 0)0. = x 0x trans. of product = product of trans. in reverse order transpose of transpose is the original matrix. X 0 = xx 0 (d) suppose that the inverse (x 0x)(cid:0)1 exists. Prove that x (x 0x)(cid:0)1 x 0 is idempotent matrix. A matrix b is idempotent if bb = b. Prove that i (cid:0) b is also idempotent matrix. A square matrix b is idempotent if bb = b. We need to prove (i (cid:0) b) (i (cid:0) b) =
