Linear Algebra

Section V. Change of Basis 247(b) Repeat the prior question with one basis for V and two bases for W.2.26 (a) If two matrices are matrix-equivalent and invertible, must their inversesbe matrix-equivalent?(b) If two matrices have matrix-equivalent inverses, must the two be matrixequivalent?(c) If two matrices are square and matrix-equivalent, must their squares bematrix-equivalent?(d) If two matrices are square and have matrix-equivalent squares, must they bematrix-equivalent?̌ 2.27 Square matrices are similar if they represent the same transformation, buteach with respect to the same ending as starting basis. That is, Rep B1 ,B 1(t) issimilar to Rep B2 ,B 2(t).(a) Give a definition of matrix similarity like that of Definition 2.3.(b) Prove that similar matrices are matrix equivalent.(c) Show that similarity is an equivalence relation.(d) Show that if T is similar to ˆT then T 2 is similar to ˆT 2 , the cubes are similar,etc. Contrast with the prior exercise.(e) Prove that there are matrix equivalent matrices that are not similar.