4. Let A be a square matrix such that AAT-1. Show that AI = ±1. 4. Let A be a square matrix such that AAT-1. Show that AI = ±1.
4. Show that an arbitrary square matrix A can be written as where Ai is a symmetric matrix and A2 is a skew-symmetric matrix 4. Show that an arbitrary square matrix A can be written as where Ai is a symmetric matrix and A2 is a skew-symmetric matrix
Let A be a matrix of size m xn. Show that AAT and AT A are both square matrices (equal number of rows and columns) (10 pts) If A is mXn then A is nXm so AA must have size mXm Similarly, A" A must be nxn
2. Let A € Mn(R). (a) Show that AAT is a semipositive definite symmetric matrix and that AAT and AT A are similar. (b) Show by example that it need not be the case that AAT and ATA are similar for A E Mn(C).
Let A be a square matrix such that det(AI – A) = 212 – 6211 + 9210 (3 points) What is the size of A? (4 points) Is A invertible? Why or why not? I (3 points) How many eigenspaces does A have?
2. Let A be any matrix and let B= AAT a. Use a 2x2 matrix A, to verify that B is symmetric. b. Write one-line proof to show that B is symmetric. Do not use part a. 3. Using Gaussian Elimination, solve the homogeneous system 2x1 + x2 – 3x3 = 0 - x2 - 4x2 + 3x3 = 0 2 1 -3 oli +3707 1-4 3lol 1-4 30
(5) Let A be some 4 x 6-matrix. Explain why AAT and AT A are defined. What are the sizes of these matrices?
Find ATA and AAT for the following matrix. 4-1-4-6 31 2 -5 4 AAT = What do you observe? This answer has not been graded yet.
Let A and B be square matrices and P be an invertible matrix. If A- PBP-,show that A and B have the same determinant. Let A and B be square matrices and P be an invertible matrix. If A- PBP-,show that A and B have the same determinant.
(a) Let A be a real n x m matrix. (i) State what conditions on n and m, if any, are needed such that the matrix AAT exists. Justify your statement. (ii) Assuming that the matrix AA exists, find its size. (iii) Assuming that the matrix AAT exists, prove using index notation that all diagonal elements of AAT are positive or equal to zero. (iv) Let 12 5 -3 A= 3-4 2 Calculate (AAT) -- (show all your working). 2)
, then n lim Let Ά be a square matrix. Prove that if ρ(A)<1 Use the following fact without proof. For any square matrix A and any positive real number ε , there exists a natural matrix norm I l such that l-4 ll < ρ (d) +ε IIA" 11-0