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3. Let det(A) = 3 and det B = –2. Find the indicated determinants: (a) det(AB)...
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.
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...
1. (10 points) Let A and B be 3 x 3 matrices, with det A = -3 and det B = 2. Compute (a) det AB (6) det B4 (c) det 3B (d) det A"B" AT (e) det B-AB
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)
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 u and v be the vectors shown in the figure to the right, and suppose u and v are eigenvectors of a 2 x2 matrix A that correspond to eigenvalues -2 and 3, respectively. Let T: R2 R2 be the linear transformation given by T(x)-Ax for each x in R2, and let w-u+v. Plot the vectors T(u), T(v), and T(w). 2- u -2 2 4 -2 10- T(v) T(w -10 10 T(u) -10- Ay 10- T(v) T(w) T(u) 10...
1 01 4. Let A 11. (a) Find ATA, (b) find det (AAT). [10 points 1 2 3 0 5. Calculate the determinant of D 03 [10 points 0 2 0 7 2 1 1 6. Find the inverse of A 1 3 using the method in section 5.3. 10 points
If A and B are 3 x 3 matricies for which det A = 2, det B =-2 find the following determinants: (all entries below are either integers or proper fractions in lowest terms) det(A)- det(B-44BA) = det(4(A(B-1)) detC4BT)-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.