show that 9- a) A is orthogonal if and only if A' is orthogonal b) A is orthogonal if and only if A is orthogonal c) A& B are orthogonal then AB is orthogonal d) A is orthogonal then det(A)=1...
True or False det(AB) det(BA) det(A B) det(A) + det(B) det(CA) c det(A) = C det((AB)T) det(A) det(B) det(B) => A = B det(A) det(A) det(A) A triangular matrix is nonsingular if and only if its diagonal entries are all nonzero.
Let A and B be 3x3 matrices, with det A=9 and det B = - 6. Use properties of determinants to complete parts (a) through (e) below. a. Compute det AB. det AB = (Type an integer or a fraction.) b. Compute det 5A. det 5A = (Type an integer or a fraction.) c. Compute det BT. det BT = (Type an integer or a fraction.) d. Compute det A-7. - 1 det A (Type an integer or a simplified...
If A, B are 3 x 3 matrices such that det(AB-1) = 12 and det(A) = 4. Find 1) det(B) 2) det(AT. (3B)-1) 3) If A? + AB = { 1, find det(A + B)
3. Let det(A) = 3 and det B = –2. Find the indicated determinants: (a) det(AB) (b) det(B-1A) (c) det(AAT) (d) det(3BT)
(A) If d=gcd(a,b) and m=lcm(a,b), prove that dm=|ab|. (B) Show that lcm(a,b)=ab if and only if gcd(a,b)=1 (C) Prove that gcd(a,c)=gcd(b,c)=1 if and only if gcd(ab,c)=1 for integers a, b, and c. (Abstract Algebra)
(1 point) If A and B are 3 x 3 matrices, det(A) = -5, det(B) = 9, then det(AB) = det(-2A) = det(AT) = det(B-1) = det(B3) =
Problem 1 (20): Let a, b,c,d ER. Show that (axb) (cx d) = det(A), where A =( index notation. laid b. d) using
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove that det ((-A-t +1 where t = Tr(A). 44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove...
2. A property of determinants states, det(AB) = det(A) det(B). Let A be a singular, diagonalizable matrix. What does this property imply about the matrices P, P/, and D? Explain what this means in the context transformation matrices.
4. Let A and B be 4 x 4 matrices. Suppose det A= 4 and det(AB) = 20. (a) (4 points) What is det B? (b) (4 points) Is B invertible? Why or why not? (c) (4 points) What is det(AT)? (d) (4 points) What is det(A-1)? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of AT are linearly independent. [2] and us 6. (6 points) Let vi...