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Exercise Show that

A = [ 1 0 -2 ], B = [ -3 -2 2 ]
0 1 0 0 1 0
2 1 -3 -2 -1 1

are inverse to each other. Then compute

C = A[ 1 0 0 ]B
0 -1 0
0 0 0

and C2001, (AC)-1, A-2.

Answer By this criterion, to verify A and B are inverse to each other, it suffices to show AB = I. A straightforward computation confirms this. Another straightforward computation gives us

C = [ -3 -2 2 ]
0 -1 0
-6 -5 4

Then by C = ADB, C2001 = (ADB)(ADB)...(ADB) = AD(BA)D(BA)D...D(BA)DB = ADIDID...DIDB = AD2001B, and the obvious equality D2001 = D, we conclude that C2001 = C.

By this equality, we have

(AC)-1 = C-1A-1 = CA-1 = [ 1 0 0 ]
0 -1 0
2 -1 0

where we used C-1 = (ADB)-1 = B-1D-1A-1 = ADB = C.

Finally, we have

A-2 = B2 = [ 5 2 -4 ]
0 1 0
1 2 -3