) Use the theorem on triangular matrices, to determine the deter minant of the matrix (it...
1. Determine which of the following matrices are invertible. Use the Invertible Matrix Theorem (or other theorems) to justify why each matrix is invertible or not. Try to do as few computations as possible. (2) | 5 77 (a) 1-3 -6] [ 3 0 0 1 (c) -3 -4 0 | 8 5 -3 [ 30-37 (e) 2 0 4 [107] F-5 1 47 (d) 0 0 0 [1 4 9] ſi -3 -67 (d) 0 4 3 1-3 6...
7. Consider the Theorem: Suppose A and B are two lower triangular matrices (Defined in 8 3.1), of order n. Then, the product AB is also a lower triangular matrix. Likewise for upper triangular matrices. (We say that the set of lower triangular matrices, of order n, is closed under multiplication.) Prove this theorem, for n = 3, by multiplying the following two matri- ces: a1 0 0 A bi b 0 1 0 0 and B 2 0 21...
An Triangular matrix is a square matrix whose elements below the diagonal are defined to be 0. For example, the matrix element Mr,c = 0 if r > c. The following is an example matrix of size 4. 0 1 2 3 0 100 200 300 400 1 0 500 600 700 2 0 0 800 900 3 0 0 0 1000 While it is possible to use a regular 2D array to represent an Triangular matrix, doing so is...
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...
HW10P5 (10 points) 3 2 -1 Let A be the matrix A = 1-3 0 6 -2 1 a. (4 pts) Find the multipliers l21, 131,132 and the elemention matrices E21, E31, E32 b. (2 pts) Use the multipliers l21, 131,132 to construct the lower triangular matrix, L c. (2 pts) Use the elimination matrices to determine the upper triangular, U, matrix of A d. (2 pts) verify that LU A
Use the matrix P to determine if the matrices A and A' are similar. --( :-2). --[-20 --a). ---( -2 -9] p-1- p-1AP = Are they similar? Yes, they are similar. No, they are not similar.
2. (a) Let A be the matrix A -4 21 8 -40 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P. Use Gaussian elimination with partial pivoting to find an upper triangular matrix U, permutation matrices Pi and P2 and lower triangular matrices Mi and M2 of the form 1 0 0 Mi-1A1 10 a2 0 1 M2 0 0 0 b1 with ail...
HW10P5 (10 points) Let A be the matrix A =13 5 0 (3 pts) Find the elementary matrices that perform the following row operations in sequence: a. 21 * 2 2. E31 : R3 R1R3 b. (3 pts) Show that the elementary matrices you found in (a) can be used as elimination matrices to determine the upper triangular, U, matrix of A. (4 pts) Find the lower triangular, L, matrix that verifies A C. = LU.
Use MATLAB to write your codes Consider a matrix A with block matrices as follows: A = {A-11 A_12 0 A-22] It can be shown that the inverse of A can be calculated by inverse of submatrices if A11, and A22 are squared matrices: A^-1 = [A_11 A_12 0 A_22] = [A6-1 _11 -A^-1 _11 A_12 A^-1 _22 0 A^-1 _22 Now consider a Matrix A with following submatrices: A11 = identity matrix A22 = identity matrix A12 = [12...
linear algebra Use the matrix P to determine if the matrices A and A' are similar. P = 15 9 -20 -11 1 p-1 p-1AP = Are they similar? Yes, they are similar. No, they are not similar.