For each of the following statements about square matrices, A and B, give a proof or...
Problem 3. Determine (with proof) whether each of the following statements is true or false. (a) For every m xn matrix A, det(AAT) = det(ATA) (b) Let A be an invertible n xn matrix, and suppose that B, C, and D are n x n matrices [det(A) |det(C) det (B) CA-1B. Then the 2 x 2 matrix is not invertible satisfying D (c) If A is an invertible n x n matrix such that A = A-1 then det(A) =...
Algebra of matrices. 3. (a) If A is a square matrix, what does it mean to say that B is an inverse of A (b) Define AT. Give a proof that if A has an inverse, then so does AT. (c) Let A be a 3 x 3 matrix that can be transformed into the identity matrix by perform ing the following three row operations in the given order: R2 x 3, Ri R3, R3+2R1 (i) Write down the elementary...
: [3 marks] Let A and B be 3 x 3 matrices. Consider the following statements. (1) If det(A) = 1 then det(24-1) = 2 (11) det(I + A) 3+det() (111) det(A + BT) = det(B+ 4) = Determine which of the above statements are True (1) or False (2). So, for example, if you think that the answers, in the above order, are True False False, then you would enter '1.2.2' into the answer box below (without the quotes).
26. This exercise and Exercises 27 and 28 give Dirichlet's beautiful proof that 2 is a biquadratic residue modulo p iff p can be written in the form A 64B', where A, Be Z. Suppose that p1 (4). Thenp b by Exercise 24. Take a to be odd. Prove the following statements (a) (a/p)1 b) (a b)lp) a (c) (a b) 2ab (p. (d) (a b(2ab)4(p). [Hint: 2p (a + b)* + (a bJ p-1V2 P14 26. This exercise and...
Is Wa subspace of R2x2? Give a detailed general proof if your answer is yes or a specific counter example if your answer is no. W = {A € R2x2 | det(A) = 0}
Is Wa subspace of R2x2? Give a detailed general proof if your answer is yes or a specific counter example if your answer is no. W={A € R2*2 | det(A) = 0}
Verify the following properties, using any distinct, invertible A, B, 4×4 upper triangular matrices of your choice: 3. (0.5 marks each) Verify the following properties, using any distinct, invertible A, B, 4 x 4 upper triangular matrices of your choice: (a) The inverse of an upper triangular matrix is upper triangular; (b) (AB)- B-1A-1 (e) trace(AB) trace(BA); (d) det(AB) det (BA) example of matrices A, B such that det(AB) det(BA) (BONUS 1 mark) Give an 3. (0.5 marks each) Verify...
4. Assume that A, B E Mnxn(R). Prove or disprove each of the following statements. (a) If AB is a product of elementary matrices, then A is a product of elementary matrices. (b) If R is the RREF of A, then det A = det R. (c) If det A-det B, then A = B.
1. (20pts) Prove or disprove each of the following statements. If true, then write a proof for the statement. If false, then give a specific explicit example. a) {12a + 4b: a and b are integers} = {4c: c is an integer), and b) For sets A, B and C: A(BUC)=(A\B)U(A\C).
(E) In general, do each of the following two statements hold? If yes, proof the result you may refer to the literature if you also add the citation. If the result does not hold, give a counter example. Answering part (c) of the question will assist you with answering part (a). (a) E(X) =E(X) (c) Let X be the outcome of the throw of a die. i. Compute E(X) ii. Now compute E(1/X). iii. Now compute 1/E(X) iv. Compare your...