Question

设矩阵A,B均为实正交矩阵且|A|=-1,|B|=1,试证明:|A+B|=0

Answer 1

因为 A,B是正交矩阵

所以 AA^T=A^TA=E,BB^T=B^TB=E

所以有

|A+B|

= |(A+B)^T|

= |A^T+B^T|

= - |A||A^T+B^T||B|

= - |AA^TB+AB^TB|

= - |B+A|

= - |A+B|

所以 |A+B| = 0.