linear algbra 2. Let A, B be the following 3 x 3 matrices: A= C? 1...
linear algebra 3. Let A be the following matrix: A= 0 -2 6 0 0 C 6 C 02 0 0 8 0 0 5 T 3 -1 7 6 2 - 4 04 (a) Find det(A). Show your work Express your answer in terms of x. (b) Identify the value(s) of x for Nul (A) = {0}.
linear algebra 1. Suppose that A, B, C, D, E are matrices of the following sizes: А B с D E (4 x 5) (4 x 4) (5 x 2) (4 x 2) (5 x 4) Determine whether the expression is defined and, if so, the size of the resulting matrix. (a) EAC (b) AC + D (c) E(5B + A) Bi and S(T) = Dr. Is the (d) Suppose T, S are matrix transformations defined by T(T) composition To...
linear algebra 1 2. Let A be the 3 x 3 matrix: A= 3 3 0 -4 1-3 5 1 (a) Find det(A) by hand. (b) What can you say about the solution(s) to the linear system Az = ? A. No Solutions B. Unique Solution C. Infinitely Many Solutions (c) Is A invertible?
2. Given that u., and ware three solutions of the linear system Az = b. Verify that the vector cu+du+ (1-c-d)w is also a solution of Ar = b for any scalars DER - 2 1 1 Let A = 1 1 - 2 1 Determine whether the system Az = b is consistent for every beR. 1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only...
3. optel Let 2 2 2 4 1 3 3 1 1 (a) Determine whether the product AB is defined. If yes, determine whether it is invertible. (b) Determine whether the product ABC is defined. If yes, determine whether it is invertible [epta Consider the system of ne linear equations in n unknowns given by As - b, where A € Mn,n is the n x n matrix of coefficients, as is the vector of unknowns and b E R"...
3 2 -1 1 Determine whether AT B is invertible Given the matrices A = [ 2 -4and B = -1 or not, where AT denotes the transpose of matrix A. 5. 1 -3 2
linear algebra 1. Consider the following matrices 01 and B=[3 0 4 3 A=[-1 2 O Show that (BA) A-1B-1
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...
A. In each case, find the matrix A T [2 1 1-201)=10 5 Problem 4. a. 10.5p (A+5 B. Let A and B denote n × n invertible matrices. a. 10.5pl Show that A-1 B-1A(A+ B)B-1. a. [0.5p] İf A+ B is also invertible, show that A-1-B-1 is invertible and find a formula for (AB A. In each case, find the matrix A T [2 1 1-201)=10 5 Problem 4. a. 10.5p (A+5 B. Let A and B denote n...
Let A, B, C, and D be matrices with the following sizes: A, 5×3 B, 3×2 C, 3×5 D, 1×3 Which of the following matrix operations are defined? i) AB (ii) A + 1 4 C (iii) DC