An invertible square matrix A satisfies A^3 +3A^2 −25A+21I = O, where I and O are the identity and zero matrices, respectively.
Find the inverse of A^2
An invertible square matrix A satisfies A^3 +3A^2 −25A+21I = O, where I and O are...
Problem 3. Give the definitions of an invertible square matrix and of the inverse of Let A be a square matrix. List at least five conditions that are equivalent to A being Prove that the inverse of a square matrix is unique if it exists. a square matrix. invertible. Problem 3. Give the definitions of an invertible square matrix and of the inverse of Let A be a square matrix. List at least five conditions that are equivalent to A...
3. (10 points) Simultaneous left inverse The two matrices 3 2] and both left-invertible, and have multiple left inverses. Do they have a common left inverse? Explain how to find a 2 × 4 matrix C that satisfies CA-CB-1, or determine that no such matrix exists. (You can use numerical computing to find C.) Hint. Set up a set of linear equations for the entries of C. Remark. There is nothing special about the particular entries of the two matrices...
Algebra of matrices. 3. (a) If A is a square matrix, what does it mean to say that B is an inverse of A (b) Define AT. Give a proof that if A has an inverse, then so does AT. (c) Let A be a 3 x 3 matrix that can be transformed into the identity matrix by perform ing the following three row operations in the given order: R2 x 3, Ri R3, R3+2R1 (i) Write down the elementary...
2. Inverse of a square matrix: Determine the inverse matrix [A™'] of the given square matrix [A] using the Gauss-Jordan Elimination Method (GEM), and verify that [A-!] [A] = I where I is the identity matrix. A = [ 1 4 -27 0 -3 -2 | -3 4 1
3. (20 %) If a square matrix A satisfies (20201 – A)3 = 0, is it possible that A is not invertible? Explain your answer.
a) Let I be the n x n identity matrix and let O be the n × n zero matrix . Suppose A is an n × n matrix such that A3 = 0. Show that I + A is invertible and that (I + A)-1 = I – A+ A2. b) Let B and C be n x n matrices. Assume that the product BC is invertible. Show that B and C are both invertible.
(a) Show that if the matrix B is invertible, then the only solution of the equation BX = 0 (where 0 is the zero square matrix of the same size as B) is X-0. (b) Consider a matrix partitioned in blocks, of the form (Α (в ο). с) where A and C are invertible, not necessarily of the same size. Find its inverse, itself partitioned in blocks of the same size, in terms of A, B, C. Hint: one of...
(a) Show that if the matrix B is invertible, then the only solution of the equation BX = 0 (where is the zero square matrix of the same size as B) is X-0. (b) Consider a matrix partitioned in blocks, of the form A 0 ( BC where A and C are invertible, not necessarily of the same size. Find its inverse, itself partitioned in blocks of the same size, in terms of A, B, C. Hint: one of the...
Show that if A is a square matrix that satisfies the equation A2 - 2A + I = O, then A = 2I - A The equation A - 2A I = O implies that It follows that Notice ---Select--- ---Select--- the last equation means that multiplied with A is the identity, which is what we wanted to prove. -Select--
[1 2 37 1. Is the matrix 1 0 1 invertible? If yes, what is its inverse? [O 2 -1 2. A matrix is called symmetric if At = A. What can you say about the shape of a symmetric matrix? Give an example of a symmetric matrix that is not a zero matrix. 3. A matrix is called anti-symmetric if A= -A. What can you say about the shape of an anti- symmetric matrix? Give an example of an...