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2. A property of determinants states, det(AB) = det(A) det(B). Let A be a singular, diagonalizable...
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)
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...
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...
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...
11. Prove one of the following: a. Let A and B be square matrices. If det(AB) + 0, explain why B is invertible. b. Suppose A is an nxn matrix and the equation Ax = 0 has a nontrivial solution. Explain why Rank A<n.
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 (A?)? (d) (4 points) What is det(A-?)? 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. and 2 6. (6 points) Let...
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-')? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of A™ are linearly independent. and t = [ ] 6. (6...
Let A and B be n × m, and m × n matrices over F respectively. Prove that rn ) = det(In-AB) = det(I,n-BA). In det A Let A and B be n × m, and m × n matrices over F respectively. Prove that rn ) = det(In-AB) = det(I,n-BA). In det A
Let A and B be square matrices of order 3 such that |A| = 4 and |B| = 7 (1) Find |AB|. (2) Find |2A|. (3) Are A and B singular or nonsingular? Explain. (A) A and B are both singular because they both have nonzero determinants. (B) A and B are both nonsingular because they both have nonzero determinants. (C) A is singular, but B is nonsingular because |A| < |B|. (D) B is singular, but A is nonsingular...
6. For each of the following matrices A solve the eigenvalue problem. If A is diagonalizable, find a matrix P that diagonalizes A by a similarity transformation D-PlAP and the respective diagonal matrix D. If A is not diagonalizable, briefly explain why -1 4 2 (d) A-|-| 3 1 -1 2 2 -1 0 1 6 3 (a) A- (b)As|0 1 0| (c) A-1-3 0 11 -4 0 3