Let's find B−1 :
(223−1∣∣1001)−I⇔(203−4∣∣1−101)⇔(2031∣∣1410−41)−3II⇔(2001∣∣414143−41)⇔(1001∣∣814183−41)
Then
B−1=(814183−41)
AB−1=(−222−4)(814183−41)=(−41+2141−1−43−2143+1)=(41−43−4547)
Let's find (AB−1)−1 :
(AB−1)−1=(41−43−4547∣∣1001)⇔(1−3−57∣∣4004)+3I⇔(10−5−8∣∣41204)⇔(10−5−2∣∣4301)−25II⇔(100−2∣∣−273−251)⇔(1001∣∣−27−23−25−21)
Then
(AB−1)−1=(−3.5−1.5−2.5−0.5)
By the properties of the inverse matrix
(AB−1)−1=(B−1)−1A−1=BA−1
Answer: 1. (AB−1)−1=(−3.5−1.5−2.5−0.5) , 5. BA−1
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