5. Let A, B, C, D be ($5 \times 5$)-square matrices. Suppose ABCD = $I_5$.
(a) Which of the square matrices AB, BC, CD, AC, BD, AD are invertible? Justify your answer (by quoting some
appropriate theoretical result about invertibility, if necessary).
(b) Which of the equalities below are definitely true?
(1) BCDA = $I_5$.
(2) DCBA = $I_5$.
(3) DBAC = $I_5$.
(5) DABC = $I_5$.
(7) ADCB = $I_5$.
(4) ACDB = $I_5$.
(6) BDCA = $I_5$.
(8) CDAB = $I_5$.
For those equalities that you claim to be definitely true, justify your claim, with reference to the definition of
invertibility.