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Question

For 3×3 matrices A and B, which of the following statements is (are) CORRECT?
  1. AB is skew-symmetric if A is symmetric and B is skew-symmetric.
  2. (adj A)T=adj (AT) for all invertible matrix A.
  3. AB+BA is symmetric for all symmetric matrices A and B.
  4. (adj A)1=adj(A1) for all invertible matrix A.


Solution

The correct options are
B (adj A)T=adj (AT) for all invertible matrix A.
C AB+BA is symmetric for all symmetric matrices A and B.
D (adj A)1=adj(A1) for all invertible matrix A.
A is symmetric and B is skew-symmetric.
Let P=AB
Now, PT=(AB)T=BTAT=BA
PTP
AB is not skew symmetric.

We know that 
A(adj A)=|A|I3
(A(adj A))T=(|A|I3)T
(adj A)TAT=|A|I3
(adj A)T=|AT|(AT)1
(adj A)T=adj(AT) for all invertible matrix A.

Let P=AB+BA
PT=(AB+BA)T=(AB)T+(BA)T
          =BTAT+ATBT=BA+AB
          =AB+BA=P
AB+BA is symmetric for all symmetric matrices A and B.

adj(A1)=A|A|     (1)
Also, A(adj A)=|A|I3
(adj A)=|A|A1
(adj A)1=A|A|     (2)
From (1) and (2), we get
(adj A)1=adj(A1)

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