Question

3.23 True or false. justify your answer

190 LINEAR TRANSFORMATIONS 3.22 Let A be a 4 x 3 matrix and B a 3 x 4 matrix. Then AB cannot be in 3.23 Suppose that A is an invertible matrix and B is any matrix for which BA i 3.24 Suppose that A is an invertible matrix and B is any matrix for which AB is 3.25 Suppose that A and B are nxn matrices such that AB is invertible. Then Hint: Think about rank.] defined. Then the matrices BA and B need not have the same rank. defined. Then the matrices AB and B need not have the same rank. both A and B are invertible. [Hint: Compare the rank of AB with that of and of B.] 3.26 Suppose that A is an n × n matrix such that AA' = 1. Then AA-1 as well 3.27 Suppose that A is an n x n matrix that satisfies A2 +7A-1 =0. Then A is invertible. 3.28 Suppose that A is invertible and B is row equivalent to A. Then B is invertible. 3.29 Suppose that A is invertible and B is any matrix such that AB is defined. Then AB and B have the same nullspace. 3.30 Suppose that A is an n xn invertible matrix and B is any n x n matrix. Then

Answer 1

A is invertible Ba is defined A^T B^t is defined let v be a matrix we have column are abasic for the null space of A^T B^T so that A^T B^T v = 0 there (A^-1)^T A^T B^T, so that A^T B^T v = 0 then (A^-1)^T A^T B^T - V = 0 rightarrow B^T v = 0 rightarrow B^T have the same null space of A^T b^T v = 0 rightarrow B^T v = 0 rightarrow B^T have the same null space as A^T B^T hence B^T and A^T B^T have the same rank since rank (B^T) - rank (B) and rank (A^T B^T) = rank ((Ba)^T) = rank (B^T) we have rank (B) = rank (BA)