1. Let A be a real 4 × 2 matrix and B be a real 2 × 4 matrix such that
1 0 −1 0
0 1 0 −1
AB =
.
−1 0 1 0
0 −1 0 1
Find BA. [20 points]
A 1where A 1 , A 2 , B 1 , B 2 are 2 × 2 matrices. Then
Solution. Let A =
and B = B 1 B 2
A 2 A 1 B 1 A 1 B 2=
B 1 B 2
A 2 B 1 A 2 B 2therefore, A 1 B 1 = A 2 B 2 = I 2 and A 1 B 2 = A 2 B 1 = −I 2 . Then B 1 = A −1 1 , B 2 = −A −1 1 and A 2 = B −1 2 =
−A 1 . Finally,
2 0
BA = B 1 B 2
= B 1 A 1 + B 2 A 2 = 2I 2 =
0 2
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