3. (10 points) Show that the determinant of a matrix is unaffected by a unitary change...
Question 19 [10 points] If the determinant of the first matrix below is as given, find the determinant of the other matrix. a b c det r s t 5 x y z 2x 2y 2z a+2x b+2y c+2z Official Time: 22:24:28 SUBMIT AND MARK
(10 pts) If the determinant of a 5 x 5 matrix A is det (A) = 8, and the matrix B is obtained from A by multiplying the second column by 9, then det (B) =
Problem 8. a) Find the determinant det (A) for the matrix [1 -3 41 A 2 0 -1 1 b) Decide whether the matrix A has an inverse. If the inverse matrix A-1 exists, find its determinant det(A-1).
(12 points) Evaluate the determinant of the matrix D using cofactor expansion down the second column, then find det(3D) and det((2D)-1). D = [ 1 -5 301 3 0 4 3 -1 0 -3 0 I 3 8 6 2
5. A 3 × 3 matrix is given by A=1020 -i 0 1 (a) Verify that A is hermitian. (b) Calculate Tr (A) and det (A), where det (A) represents the determinant of A (c) Find the eigenvalues of A. Check that their product and sum are consistent with Prob. (5b) (d) Write down the diagonalized version of A (e) Find the three orthonormal eigenvectors of A. (f) Construct the unitary matrix U that diagonalizes A, and show explicitly that...
5. (10 points) Find the determinant of the given matrix A by using cofactor expansion. Then find the determinant of A. 1 2 A= | -2 3 3 -5 5 1 7 0 /
6. (5 points) Suppose the elementary matrix E is of this form (a) Compute the matrix multiplication EB (b) Compute the determinant of EB using the cofactor expansion along the 1st row of the matrix, and show that the determinant is equal to -det(B) (MUST use the cofactor expansion, no points will be given for other meth- ods.) Hint: Same, don't expand everything out, you will be drown in a sea of bij, you should look at the cofactor expansion...
(1 point) Find the determinant of the matrix A= -9 1-8 3 | det(A) =
tion 135 The matrix A = b c df 9 Lh j k] has a determinant of 6. 3d 29 -6f (a) What is the determinant of the matrix B = 3a 2c - 6b 3h 2k-6j] List the row & column'operations that were performed on A to produce B and state what each of these operations' does to the determinant 3d 39 3f (b) What is the determinant of the matrix C = 2a 2c 25 -6h -6 Explain...
(1 point) Compute the determinant of the matrix -1 -2 -4 -6 -7 -7 7 7 A= 0 0 0 0 -4 -5 7 det(A) (1 point) Find the determinant of the matrix 6 A- 6 -9 -7 det(A) (1 point) Find the determinant of the matrix 2 2 -2 B= 1 -1 2 3 -2 det (B)